Unit 5
MCR 3U1
Lesson 7: Trigonometric Identities
A trigonometric identity is an equation that is true for all values of the variable for which the expression
on both sides of the equation is defined.
To prove that a given trigonometric equation is an identity, both sides of the equation need to be shown to
be equivalent. This can be done by:




Simplifying the more complicated side until it is identical to the other side or manipulating both
sides to get the same expression.
Rewriting all expressions involving tangent and the reciprocal trig ratios in terms of sine and cosine.
Applying the Pythagorean Identities where appropriate.
Using a common denominator or factoring as required.
You can use the following identities to help you prove other identities.
Quotient Identities
tan  
cot 
Examples
sin 
cos
cos
sin 
Pythagorean Identities
sin 2   cos 2   1
1  sin 2   cos 2 
1  cos 2   sin 2 
Example 1: Prove the following trigonometric identities.
1  cos 2 x
 sin x
a)
sin x
c)
sin 2 x  6 sin x  9 sin x  3

sin x  3
sin 2 x  9
b)
(sin x  cos x) 2  1  2 sin x cos x