Trigonometric Identities (script)
Author: Julia Launer
Last updated: April 18, 2011
Suppose you have to simplify an expression that looks like this:
2 cos x
sin x
Or this:
– 2 cos ² x – 2 sin ² x
Or prove this:
cot x + tan x = sec x csc x
Knowing how to manipulate the basic trigonometric, Pythagorean, and double angle identities can make
these expressions much easier to simplify. Here are the fundamental identities:
Fundamental Identities
tan x =
sin x
cos x
cot x =
1
cos x
=
tan x sin x
sec x =
1
cos x
csc x =
1
sin x
Pythagorean Identities
sin ² x + cos ² x = 1
1 + tan ² x = sec ² x
1 + cot ² x = csc ² x
Double Angle Identities
sin 2x = 2 sin x cos x
cos 2x = cos ² x – sin ² x = 2 cos ² x – 1 = 1 – 2 sin ² x
Using Identities
Simplify:
1.
2 cos x
= 2 cot x
sin x
2. – 2 cos ² x – 2 sin ² x = -2 (cos ²x + sin ²x) = - 2
Prove each identity by changing ONLY the left side of the equation until it matches the right side.
1. (1 – cos ² x) sec ² x = tan ² x
sin ² x sec ² x = tan ² x
sin ² x
sin 2 x
cos 2 x
1
= tan ² x
cos 2 x
= tan ² x
Avery Point Academic Center
Trigonometric Identities (script)
Author: Julia Launer
Last updated: April 18, 2011
2. cot x + tan x = sec x csc x
cos x
sin x
+
= csc x sec x
sin x
cos x
cos x  cos x  sin x  sin x 

+

 = csc x sec x
sin x  cos x  cos x  sin x 
cos 2 x
sin 2 x
+
= csc x sec x
sin x cos x sin x cos x
cos 2 x  sin 2 x
= csc x sec x
sin x cos x
1
= csc x sec x
sin x cos x
1
1
= csc x sec x
sin x cos x
csc x sec x = csc x sec x
3. tan x (
tan x cot x
+
) = sec x
sec x sec x
tan 2 x
+
sec x
1
tan x = sec x
sec x
tan x
tan 2 x  1
= sec x
sec x
sec 2 x
= sec x
sec x
sec x = sec x
Avery Point Academic Center