Verify/prove: (1-cos^2x)(1+cos^2x) = 2sin^2x-sin^4x

Question

I chose to manipulate the left hand side. So i got 1+cos²x-cos²x-cos⁴x

when i simplify it will be 1-cos⁴x i will break it into pieces so (1-cos²x)(1+cos2x) i can change the first term to sin so (sin²x)(1+cos²x). Nowww im lost. Please help!!!

Answer 1

The first parenthesis on the left-hand side of the equation is just sin²[x]. Since cos²[x] = 1 - sin²[x] , plug that into the second parenthesis on the left-hand side of the equation:

 

(sin²[x])(1 + (1 - sin²[x])) = 2sin²[x] - sin⁴[x]