The document discusses analytic trigonometry. It defines trigonometric functions using the unit circle, where the x-coordinate is cos(θ) and the y-coordinate is sin(θ) at an angle θ. It presents several important trigonometric identities relating functions, including the division identities, reciprocal identities, and square-sum identities. These identities are summarized in a trigonometric hexagram for recalling relationships between functions.

1. Analytic Trigonometry

2. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().

(1,0)
The unit circle x2 + y2 = 1
(x, y)

3. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
The unit circle x2 + y2 = 1
(x, y)

4. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
The unit circle x2 + y2 = 1
(x, y)

5. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
cot()
The unit circle x2 + y2 = 1
(x, y)

6. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
cot()
The following reciprocal
functions are useful in
science and engineering:
csc() = 1/sin() = 1/y,
the “cosecant of ".
sec() = 1/cos() = 1/x,
the "secant of "
(x, y)
The unit circle x2 + y2 = 1

7. Analytic Trigonometry
Let (x, y) be a point on the unit circle,
then x = cos() and y = sin().
Here is the picture of the
geometric measurements,
related to the unit circle,
of the trig–functions.

cos()
sin()
(1,0)
tan()
cot()
The following reciprocal
functions are useful in
science and engineering:
csc() = 1/sin() = 1/y,
the “cosecant of ".
sec() = 1/cos() = 1/x,
the "secant of "
Since y2 + x2 = 1, we’ve sin2() + cos2() = 1
or s()2 + c()2 = 1, or simply
that s2 + c2 = 1 all angles .
(x, y)
The unit circle x2 + y2 = 1

8. Analytic Trigonometry
An identity is an equation where every number
where the equation is defined, is a solution.

9. Analytic Trigonometry
An identity is an equation where every number
where the equation is defined, is a solution.
For example, the equation “ ” is an identity.
1
𝑥
=
𝑥
𝑥2

10. Analytic Trigonometry
An identity is an equation where every number
where the equation is defined, is a solution.
For example, the equation “ ” is an identity.
Its solutions are all numbers except for x = 0
where it´s UDF.
1
𝑥
=
𝑥
𝑥2

11. Analytic Trigonometry
An identity is an equation where every number
where the equation is defined, is a solution.
For example, the equation “ ” is an identity.
Its solutions are all numbers except for x = 0
where it´s UDF.
The identity
sin2() + cos2() = 1
is the most important
identity in trigonometry.
1
𝑥
=
𝑥
𝑥2

x=cos()
y = sin()
(1,0)
(x, y)
The unit circle
y2 + x2 = 1
sin2() + cos2() = 1

12. Analytic Trigonometry
An identity is an equation where every number
where the equation is defined, is a solution.
For example, the equation “ ” is an identity.
Its solutions are all numbers except for x = 0
where it´s UDF.

x=cos()
y = sin()
(1,0)
(x, y)
The unit circle
y2 + x2 = 1
sin2() + cos2() = 1
The identity
sin2() + cos2() = 1
is the most important
identity in trigonometry.
1
𝑥
=
𝑥
𝑥2
Note that we have the
following equivalent versions
of this identity:
sin2() = 1 – cos2()
cos2() = 1 – sin2()

13. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().

14. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().

15. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().
The Trig–HexagramThe hexagram shown here
is a useful visual tool for
recalling various relations.

16. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().
The Trig–HexagramThe hexagram shown here
is a useful visual tool for
recalling various relations.
The right side of the hexagram
is the “co”–side listing all the
co–functions.
The “co”–side

17. Analytic Trigonometry
If we divide the identity s2 + c2 = 1 by s2, we obtain
1 + (c/s)2 = (1/s)2 or that 1 + cot2() = csc2().
If we divide the identity s2 + c2 = 1 by c2, we obtain
(s/c)2 + 1 = (1/c)2 or that tan2() + 1 = sec2().
The Trig–HexagramThe hexagram shown here
is a useful visual tool for
recalling various relations.
The right side of the hexagram
is the “co”–side listing all the
co–functions.
The following three groups
of formulas extracted from this
hexagram are referred to as
the fundamental trig–identities.
The “co”–side

18. Analytic Trigonometry
The Division Identities

19. Analytic Trigonometry
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II

20. Analytic Trigonometry
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) =
sec(A) =
sin(A)cot(A) =

21. I
II III
Analytic Trigonometry
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) =
sin(A)cot(A) =
÷

22. I II
III
Analytic Trigonometry
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) =
÷

23. I II
III
Analytic Trigonometry
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

24. Analytic Trigonometry
The Reciprocal Identities
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

25. Analytic Trigonometry
The Reciprocal Identities
Start from any function, going across diagonally,
we always have I = 1 / III.
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II.
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

26. I
III
Analytic Trigonometry
The Reciprocal Identities
Start from any function, going across diagonally,
we always have I = 1 / III.
Example B: sec(A) = 1/cos(A),
÷II
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II.
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)

27. IIII
Analytic Trigonometry
The Reciprocal Identities
Start from any function, going across diagonally,
we always have I = 1 / III.
Example B. sec(A) = 1/cos(A), cot(A) = 1/tan(A)
The Division Identities
Starting from any function I,
going around the perimeter
to functions II and III,
then I = II / III or I * III = II.
Example A.
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)
÷ II

28. Square–Sum Identities
Analytic Trigonometry

29. Square–Sum Identities
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
Analytic Trigonometry

30. Square–Sum Identities
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
Analytic Trigonometry

31. Square–Sum Identities
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
tan2(A) + 1 = sec2(A)
Analytic Trigonometry

32. Square–Sum Identities
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Analytic Trigonometry

33. Square–Sum Identities
For each of the three inverted
triangles, the sum of the
squares of the top two is the
square of the bottom one.
sin2(A) + cos2(A) = 1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Analytic Trigonometry
These hexagram-identities are
the fundamental trig-identities
and we list them below.

34. Fundamental Identities
C(A) = cos(A)
S(A) = sin(A)
T(A) = tan(A)

35. Fundamental Identities
sec(A) = 1/C
csc(A) = 1/S
cot(A) = 1/T
C(A) = cos(A)
S(A) = sin(A)
T(A) = tan(A)
sec(A)C(A) = 1
csc(A)S(A) = 1
cot(A)T(A) = 1
Reciprocal Identities

36. Fundamental Identities
sec(A) = 1/C
csc(A) = 1/S
cot(A) = 1/T
Division Identities
T(A) = S/C
cot(A) = C/S
C(A) = cos(A)
S(A) = sin(A)
T(A) = tan(A)
sec(A)C(A) = 1
csc(A)S(A) = 1
cot(A)T(A) = 1
Reciprocal Identities

37. Fundamental Identities
sec(A) = 1/C
csc(A) = 1/S
cot(A) = 1/T
Division Identities
T(A) = S/C
cot(A) = C/S
Square–Sum Identities
S2(A) + C2(A) = 1
T2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
C(A) = cos(A)
S(A) = sin(A)
T(A) = tan(A)
sec(A)C(A) = 1
csc(A)S(A) = 1
cot(A)T(A) = 1
Reciprocal Identities

38. Fundamental Identities
sec(A) = 1/C
csc(A) = 1/S
cot(A) = 1/T
Division Identities
T(A) = S/C
cot(A) = C/S
Square–Sum Identities
S2(A) + C2(A) = 1
T2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Two angles are complementary if their sum is 90o.
Let A and B = 90 – A be complementary angles,
we have the following co–relations.
S(A) = C(B) = C(90 – A)
T(A) = cot(B) = cot(90 – A)
sec(A) = csc(B) = csc(90 – A)
A
A and B are complementary
C(A) = cos(A)
S(A) = sin(A)
T(A) = tan(A)
B = 90 – A
sec(A)C(A) = 1
csc(A)S(A) = 1
cot(A)T(A) = 1
Reciprocal Identities

39. cos(–A) = cos(A), sin(–A) = – sin(A)
The Negative Angle Relations
Fundamental Identities
1
A
–A
1
A
–A
sin(A)
sin(–A)

40. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))

41. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))

42. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))

43. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))
Example D. Simplify (sec(x) – 1)(sec(x) + 1)
And express the answer in sine and cosine.

44. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))
Example D. Simplify (sec(x) – 1)(sec(x) + 1)
And express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1) = sec2(x) – 1

45. Notes on Square–Sum Identities
The square–sum identities may be rearranged into
the difference-of-squares identities which have
factored versions of the formulas.
Example C.
a. 1 – S2(A) = C2(A) = (1 – S(A))(1 + S(A))
b. sec2(A) – tan2(A) = 1
= (sec(A) – tan(A))(sec(A) + tan(A))
c. cot2(A) – csc2(A) = –1
= (cot(A) – csc(A))(cot(A) + csc(A))
Example D. Simplify (sec(x) – 1)(sec(x) + 1)
And express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1) = sec2(x) – 1 = T2(x) = S2(x)
C2(x)

46. Verifying Identities
Another type of trig-algebra problems is to verify a
given equation is actually an identity.
To verify an equations is an identity we may transform
the equation with all the usual algebraic steps of
+, – , * and / both sides by the same quantity.
The goal is to manipulate it into a fundamental identity
thus justifying it’s an identity.
Example D. Verify the following identity.
a. (1 – sin(x))(1 + sin(x)) = 1/(1 + tan2(x))
(1 – sin(x))(1 + sin(x)) = 1/(1 + tan2(x)) or
?
Starting with
1 – sin2(x) = 1/(1 + tan2(x)) or
cos2(x) = 1/sec2(x) which is true.
?
Hence the equation is an identity.
?

47. Verifying Identities
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.

48. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1

49. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1
1
2sec2(x) =
1 + sin(x) 1 – sin(x)
+ 1
Multiply both sides by the LCD
?

50. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1
(
1
2sec2(x) =
1 + sin(x) 1 – sin(x)
+ 1
* (1 + sin(x))(1 – sin(x)))
Multiply both sides by the LCD
?

51. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1
(
1
2sec2(x) =
1 + sin(x) 1 – sin(x)
+ 1
* (1 + sin(x))(1 – sin(x)))
Multiply both sides by the LCD
(1 + sin(x))(1 – sin(x)) 2sec2(x) = (1 – sin(x)) + (1 + sin(x))
?
?

52. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1
(
1
2sec2(x) =
1 + sin(x) 1 – sin(x)
+ 1
* (1 + sin(x))(1 – sin(x)))
Multiply both sides by the LCD
(1 + sin(x))(1 – sin(x)) 2sec2(x) = (1 – sin(x)) + (1 + sin(x))
?
?
cos2(x)

53. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1
(
1
2sec2(x) =
1 + sin(x) 1 – sin(x)
+ 1
* (1 + sin(x))(1 – sin(x)))
Multiply both sides by the LCD
(1 + sin(x))(1 – sin(x)) 2sec2(x) = (1 – sin(x)) + (1 + sin(x))
?
?
cos2(x) 2sec2(x) cos2(x) = 2
?

54. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1
(
1
2sec2(x) =
1 + sin(x) 1 – sin(x)
+ 1
* (1 + sin(x))(1 – sin(x)))
Multiply both sides by the LCD
(1 + sin(x))(1 – sin(x)) 2sec2(x) = (1 – sin(x)) + (1 + sin(x))
?
?
cos2(x) 2sec2(x) cos2(x) = 2
?
2 = 2 which is an identity.

55. Verifying Identities
Example D. b. Verify the following identity.
1
2sec2(x) =
1 + sin(x)
The general strategy in verifying identities is to clear all
the denominators in the problem first,
then simplify the two sides.
1 – sin(x)
+
1
(
1
2sec2(x) =
1 + sin(x) 1 – sin(x)
+ 1
* (1 + sin(x))(1 – sin(x)))
Multiply both sides by the LCD
(1 + sin(x))(1 – sin(x)) 2sec2(x) = (1 – sin(x)) + (1 + sin(x))
?
?
cos2(x) 2sec2(x) cos2(x) = 2
?
2 = 2 which is an identity.
Hence the equation is an identity.

56. Verifying Identities
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.

57. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)

58. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
Clear the denominator.
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)

59. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
Clear the denominator.
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1 = (1 – 2sin2(A))(cot2(A) + 1)

60. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
Clear the denominator.
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1 = (1 – 2sin2(A))(cot2(A) + 1)
Convert to S and C. C2
S2
– 1 = (1 – 2S2)( + 1)C2
S2

61. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
Clear the denominator.
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1 = (1 – 2sin2(A))(cot2(A) + 1)
Convert to S and C. C2
+ 1 – 2C2 – 2S2
S2
– 1 = (1 – 2S2)( + 1), expandC2
S2
C2
S2
– 1 = C2
S2

62. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
Clear the denominator.
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1 = (1 – 2sin2(A))(cot2(A) + 1)
Convert to S and C. C2
+ 1 – 2C2 – 2S2
S2
– 1 = (1 – 2S2)( + 1), expandC2
S2
C2
S2
– 1 = C2
S2
–2

63. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
Clear the denominator.
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1 = (1 – 2sin2(A))(cot2(A) + 1)
Convert to S and C. C2
+ 1 – 2C2 – 2S2
S2
– 1 = (1 – 2S2)( + 1), expandC2
S2
C2
S2
– 1 = C2
S2
–2
C2
S2
– 1 = C2
S2 – 1 which is an identity.

64. Verifying Identities
Example D. c. Verify the following identity.
One way to reduce a problem to two variables is to
write all the trig–functions in sines S and cosines C.
cot2(A) – 1
=
Clear the denominator.
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1
=
cot2(A) + 1
1 – 2sin2(A)
cot2(A) – 1 = (1 – 2sin2(A))(cot2(A) + 1)
Convert to S and C. C2
+ 1 – 2C2 – 2S2
S2
– 1 = (1 – 2S2)( + 1), expandC2
S2
C2
S2
– 1 = C2
S2
–2
C2
S2
– 1 = C2
S2 – 1 which is an identity.
Hence the equation is an identity.

65. Trig –Equations
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

66. Trig –Equations
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

67. Trig –Equations
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

68. Trig –Equations
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

69. Trig –Equations
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½
or that S = 1,
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

70. Trig –Equations
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1,
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

71. Trig –Equations
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1, so x = π/2.
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

72. Trig –Equations
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1, so x = π/2.
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
By adding 2nπ where n is an integer, to the solutions
from a. we obtain all the solutions for x.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

73. Trig –Equations
Writing S(x) as S and replacing C2 as 1 – S2, we’ve
S2 – (1 – S2) = S or
2S2 – S – 1 = 0 so
(2S + 1)(S – 1) = 0
Hence 2S = 1, or S(x) = ½ so x = π/6 or 5π/6,
or that S = 1, so x = π/2.
Example E. Solve the equation S2(x) – C2(x) = S(x).
a. Find all solutions x between 0 and 2π.
b. Find all solutions for x.
By adding 2nπ where n is an integer, to the solutions
from a. we obtain all the solutions for x. Here is a list:
{π/6+2nπ, 5π/6+2nπ, or π/2+2nπ where n is an integer}.
Most trig–equations are not identities.
Hence they have specific solutions or no solution at all.

74. Fundamental Identities
sec(A) = 1/C
csc(A) = 1/S
cot(A) = 1/T
Division Identities
T(A) = S/C
cot(A) = C/S
Square–Sum Identities
S2(A) + C2(A) = 1
T2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
Two angles are complementary if their sum is 90o.
Let A and B = 90 – A be complementary angles,
we have the following co–relations.
S(A) = C(B) = C(90 – A)
T(A) = cot(B) = cot(90 – A)
sec(A) = csc(B) = csc(90 – A)
A
A and B are complementary
C(A) = cos(A)
S(A) = sin(A)
T(A) = tan(A)
B = 90 – A
sec(A)C(A) = 1
csc(A)S(A) = 1
cot(A)T(A) = 1
Reciprocal Identities

75. The Negative Angle Relations
Fundamental Identities
cos(–A) = cos(A), sin(–A) = – sin(A)
1
A
–A
1
A
–A
sin(A)
sin(–A)
+
+ +

76. Analytic Trig
Exercise. A. Simplify by replacing all the
expressions to sine, cosine and constants.
1. cot(A)tan(A)
17. csc(A)sec(A)/cot(A)
2. cos(A)tan(A)
15. –2sin2(A) – 2cos2(A)
3. cot(A)sec(A) 4. cos(A)tan(A)
5. cos2(A)tan(A) 6. cos(A)tan2(A)
7. cot(A)tan2(A) 8. cot2 (A)tan (A)
9. (ccs(x) – 1)(csc(x) + 1) 10. (1 – tan(A))(tan(A) + 1)
11. (cos(x) – 1)(cos(x) + 1) 12. (csc(x) – cot(A))(csc(x) + cot(A))
13. (1 – sin(A))(1 + sin(A)) 14. (tan(x) – sec(A))(cot(x) + sec(A))
16. 2 – 2cot2(A)
18. cot(A)sec(A)/csc(A)
19. 20.1
1 – cos(x) +
1
1 + cos(x)
1
1 – csc(x) +
1
1 + csc(x)

77. Analytic Trig
Exercise. B. Verify the following identities.
1. 2.1 − cos2 𝑥
sin 𝑥
= sin 𝑥 cot 𝑥 tan 𝑥 = sec 𝑥 csc 𝑥
3. cot 𝑥
csc 𝑥
= cos 𝑥 4. sin2 𝑥 1 + cot2 𝑥 = 1
5. 6.sin2
𝑥
cos 𝑥
= sec 𝑥 − cos 𝑥
sin2 𝑥 + tan2 𝑥 + cos2 𝑥 = sec2 𝑥
7.
1 − cos 𝑥
1 + cos 𝑥
= cot 𝑥 − csc 𝑥 2 8. tan2 𝑥
cos 𝑥 + 1
=
sec 𝑥 − 1
cos 𝑥
9.
tan2
𝑥
cos 𝑥 + 1
=
sec 𝑥 − 1
cos 𝑥
10. csc(x) + cot 𝑥
tan 𝑥 + sin 𝑥
≡ cot 𝑥 csc(𝑥)
11.
1 + cos 𝑥
1 − cos 𝑥
−
1 − cos 𝑥
1 + cos 𝑥
= 4 cot 𝑥 csc 𝑥
12. sec4 𝑥 − sec2 𝑥 = tan4 𝑥 + tan2 𝑥
13. tan(x) sin2 𝑥 = tan2 𝑥 + cos2 𝑥 − 1
14.
15.
cot 𝑥 + 1
cot 𝑥 − 1
=
tan 𝑥 + 1
1 − tan 𝑥
sin 𝑥 + cos 𝑥 =
sin 𝑥
1 − cot 𝑥
+
cos 𝑥
1 − tan 𝑥

78. Analytic Trig
Exercise. C.
Solve the following futons. equation S2(x) – C2(x) = S(x).
Find the solutions x between 0 and 2π,
then list all the solutions.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
sin2 𝑥 − 1 = 0 4 cos2 𝑥 − 1 = 0
tan2 𝑥 − 3 = 0 cot2 𝑥 − 1 = 0
2sin2 𝑥 + sin (x) = 0 cos2 𝑥 − cos (x) = 0
tan 𝑥 = 1 cos 𝑥 = −
2
2
sin 𝑥 = −
1
2
sec 𝑥 = −2

79. Analytic Trig
Answers. A.
1. 1 3. 1/S 5. CS
7. S/C 9. S2/C2 11. –S2(A)
13.C2(A) 15. 1/C2 19. 2/S2
9.
C. 1.
𝜋
4
+ 𝑛𝜋 3.
7
6
𝜋 + 2𝑛𝜋,
11
6
𝜋 + 2𝑛𝜋
5.
𝜋
2
+ 𝑛𝜋 7.
𝜋
3
+ 2𝑛𝜋,
2𝜋
3
+ 2𝑛𝜋
𝑛𝜋,
7
6
𝜋 + 2𝑛𝜋,
11
6
𝜋 + 2𝑛𝜋