Trigonometric Identites

Question

How do you verify the identity of sin^2x-sin^4x=cos^2x-cos^4x

Answer 1

Relatively easy.  Let's work on the left side, only.

 

Factor out sin²x:   sin²x(1-sin²x) = sin²xcos²x = cos²x(1-cos²x) = cos²x - cos⁴x...and this latter equivalent is identical to the right side, so it's done (verified).  

 

Note that I made liberal use of the commutative multiplication property and of the basic Pythagorean identity.

Answer 2

the best way is generally to convert everything to a single function.  For some reason I chose cosine.

 

sin²x - sin⁴x = cos²x - cos⁴x

 

(1- cos²x) - (1 - cos²x)² = cos²x (1 - cos²x)

 

(1 - cos²x) [(1) - (1-cos²x)] = cos²x (1- cos²x)

 

(1 - cos²x) (0 + cos²x) = cos²x (1 - cos²x)