This document contains derivations of trigonometric identities. It simplifies expressions involving trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant by rewriting them in terms of other trigonometric functions or common denominators. It also uses factoring and trigonometric angle addition and subtraction formulas to derive new identities from existing ones. The document derives over a dozen new trigonometric identities through algebraic manipulation and rearrangement of trigonometric expressions.

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Section 7.4
Trigonometric Identities

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tan
(a) Simplify by rewriting each trigonometric function in terms of sine and cosine functions.
sec
sin 1 cos
(b) Show that by multiplying the numerator and denominator by 1 cos
1+cos sin
θ
θ
θ θ
θ
θ θ
−
= −
sin
tan sin coscos(a) sin
1sec cos 1
cos
θ
θ θ θθ θ
θ θ
= = × =
( )
( ) ( )
sin 1 cossin
(b)
1 cos 1 cos 1 cos
θ θθ
θ θ θ
−
=
+ + −
( )
2
sin 1 cos
1 cos
θ θ
θ
−
=
−
( )
2
sin 1 cos
sin
θ θ
θ
−
=
1 cos
sin
θ
θ
−
=

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(a) Simplify
1
1− sinu
+
1
1+ sinu
by rewriting the expression over a common denominator.
(b) Simplify
1− cos2
v
sinv + cosvsinv
by factoring.
( )
( ) ( )
( )
( ) ( )
1 sin 1 sin
1 sin
1 1
(a)
1 sin 1 si sn 1 in
u u
uu u u
+ −
+ −
+
− + 2
1 sin 1 sin
1 sin
u u
u
+ + −
=
−
=
2
cos2
u
= 2sec2
u
( ) ( )
( )
2
1 cos 1 cos1 cos
(b)
sin cos sin sin 1 cos
v vv
v v v v v
+ −−
=
+ +
1 cos 1 cos
csc cot
sin sin sin
v v
v v
v v v
−
= = − = −

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( )sin cot tan secθ θ θ θ+ =
( )
cos sin
sin cot tan sin
sin cos
θ θ
θ θ θ θ
θ θ
 
+ = + ÷
 
2 2 2
sin cos sin
cos
cos cos
θ θ θ
θ
θ θ
+
= + =
1
sec
cos
θ
θ
= =

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sin
csc cot
1 cos
θ
θ θ
θ
− =
+
1 cos
sin
θ
θ
−
=
1 cos
csc cot
sin sin
θ
θ θ
θ θ
− = −
( ) ( )
( )
1 cos 1 cos
sin 1 cos
θ θ
θ θ
− +
=
+ ( )
2
1 cos
sin 1 cos
θ
θ θ
−
=
+
( )
2
sin
sin 1 cos
θ
θ θ
=
+
sin
1 cos
θ
θ
=
+

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2
2
2
sin tan
tan
cos cot
θ θ
θ
θ θ
−
=
−
2
2
2
2
sin
sin
sin tan cos
coscos cot cos
sin
θ
θ
θ θ θ
θθ θ θ
θ
−
−
=
− −
2
2
sin cos sin
cos
sin cos cos
sin
θ θ θ
θ
θ θ θ
θ
−
=
−
2
2
sin cos sin sin
cos sin cos cos
θ θ θ θ
θ θ θ θ
−
= ×
−
( )
( )
2
2
sin sin cos 1
cos sin cos 1
θ θ θ
θ θ θ
−
=
−
2
2
2
sin
tan
cos
θ
θ
θ
= =

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cos 1 sin
1 sin cos
θ θ
θ θ
+
=
−
( )
( ) ( )
cos 1 sincos
1 sin 1 sin 1 sin
θ θθ
θ θ θ
+
=
− − +
( )
2
cos 1 sin
1 sin
θ θ
θ
+
=
−
( )
2
cos 1 sin
cos
θ θ
θ
+
=
1 sin
cos
θ
θ
+
=

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2 csc sin
cot
sin
θ θ
θ
θ
−
=
1
sin
csc sin sinsin
sin sin sin
θ
θ θ θθ
θ θ θ
 
− ÷−  
=  ÷ ÷
  ÷
 
2 2
2
2 2
1 sin cos
cot
sin sin
θ θ
θ
θ θ
−
= = =

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3 2
1 csc sin cosθ θ θ− =
3
3 sin
1 csc sin 1
sin
θ
θ θ
θ
− = −
2 2
1 sin cosθ θ= − =

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