how would you prove that

Question

(cos^2x-sin^2x)(1-cos^2xsin^2x)=
(cos^6-sin^6)?

Answer 1

On the left
cos^2 x - cos^4 x sin^2 x - sin^2 x + cos^2 x sin^4 x

= cos^2 x - cos^4 x (1 - cos^2 x) - sin^2 x + sin^4 x (1 -sin^2 x)

= cos^2 x -cos^4 x + cos^6 x - sin^2 x + sin^4 x - sin^6 x

= cos^6 x - sin^6 x
+ cos^2 x - sin^2 x + sin^4 x - cos^4 x

= cos^6 x - sin^6 x
+ cos^2 x - sin^2 x
+ (sin^2 x - cos^2 x)(sin^2 x + cos^2 x)

= cos^6 x - sin^6 x
+ cos^2 x - sin^2 x
- (cos^2 x - sin^2 x)(sin^2 x + cos^2 x)

= cos^6 x - sin^6 x
- (cos^2 x - sin^2 x)(-1 + sin^2 x + cos^2 x)
but
-1 + sin^2 x + cos^2 x = 0

Answer 2

Wow, that was a good one !