2. Basic Idenitities
Ratios
sin A = a/h
cos A = b/h
tan A = a/b
Inverse Ratios
csc A = 1/sinA = h/a
sec A = 1/cosA =h/b
cot A = 1/tanA =b/a
3. The Three Identities
1) sin2Θ + cos2Θ = 1
N.B. sin2Θ = (sinΘ)2 . That could be
confused with sinΘ2 .
2) 1 + tan2Θ = sec2Θ
N.B. sec2Θ = 1/sin2Θ
3) 1+ cot2Θ = csc2Θ
N.B. cot2Θ = 1/tan2Θ; csc2Θ = 1/sin2Θ
4. Proving Your Identity
To prove an identity, identify the most complex
side and simplify it to express it in the same terms
as the other side. To do so, try the following:
Substitute on or more basic identities to simplify
the expression.
Perform operations or factor to find a basic
identity or a factor common to the numerator and
denominator.
Multiply the numerator and the denominator by
the same trigonometric expression.
Express the various functions using sine and
cosine functions.
5. Example of Proof: 1/sec2Θ + 1/csc2Θ = 1
It helps to divide the equation into a
Left Hand Side (LHS) & a Right Hand Side RHS).
1/sec2Θ + 1/csc2Θ
cos2Θ + sin2Θ
1 (Trig Identity)
So 1
Proven/QED
Quad Erat Demonstratum
It has been shown.
1
= 1
6. Example of Proof: tan2x - sin2x = sin2xtan2x
tan2x - sin2x
sin2x/cos2x - sin2x
Factor sin2x out
Sin2x (1/cos2x -1)
Sin2x (sec2x -1)
Check identities
Sec2x = 1 + tan2x
So, sec2x – 1 = tan2x
Sin2x(tan2x)
sin2x tan2x
= sin2x tan2x
QED
7. Example of Proof: Last one
2cos2t-cost-1
cos t -1
Numerator: Quadratic expression
(2cos t + 1)(cos t -1)
(cos t -1)
Cancel out common factors
2 cos t + 1
Remember: sec t = 1/cos t
So cos t = 1/sec t
2 ( 1 ) + 1
(sec t)
Common Denominator
2 ( 1 ) + sec t = 2 + sec t
(sec t) sec t sec t
sec t + 2
sec t
= sec t + 2 QED
sec t
8. Exam Question
Prove that,
x
x
x
x
xx 2
2
2
2
22
tan
cos
cos1
1sec
1tansin
Show your work.
9. Exam Question
For all values of A (for which A is defined), the expression tan A + cot A is equal to
A) sin A cos A. C) sec A cosec A.
B) sec A cos A. D) sin A cosec A.
