SOLUTION: Please help me with the following question. Prove this identity: (1-cos^2x)(1+cos^2x) = 2sin^2x - sin^4x

Algebra ->  Trigonometry-basics -> SOLUTION: Please help me with the following question. Prove this identity: (1-cos^2x)(1+cos^2x) = 2sin^2x - sin^4x      Log On


   

Question 1058611: Please help me with the following question.
Prove this identity:
(1-cos^2x)(1+cos^2x) = 2sin^2x - sin^4x
Found 3 solutions by Alan3354, josgarithmetic, MathTherapy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
(1-cos^2x)(1+cos^2x) = 2sin^2x - sin^4x
sin^2*(1+cos^2x) = 2sin^2x - sin^4x
1+cos^2x = 2 - sin^2x
cos^2(x) = 1 - sin^2(x) QED

Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
(1-cos^2x)(1+cos^2x) = 2sin^2x - sin^4x
%281-cos%5E2x%29%281%2Bcos%5E2x%29+=+2sin%5E2x+-+sin%5E4x
%281-cos%5E2%28x%29%29%281%2Bcos%5E2%28x%29%29+=+2sin%5E2%28x%29+-+sin%5E4%28x%29

RHS,
sin%5E2%28x%29%282-sin%5E2%28x%29%29
%281-cos%5E2%28x%29%29%282-sin%5E2%28x%29%29
%281-cos%5E2%28x%29%29%282-%281-cos%5E2%28x%29%29%29
%281-cos%5E2%28x%29%29%282-1%2Bcos%5E2%28x%29%29
%281-cos%5E2%28x%29%29%281%2Bcos%5E2%28x%29%29------------Identical to LHS

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Please help me with the following question.
Prove this identity:
(1-cos^2x)(1+cos^2x) = 2sin^2x - sin^4x
LHS:%281+-+cos%5E2+%28x%29%29%281+%2B+cos%5E2+%28x%29%29+=+2+sin%5E2+%28x%29+-+sin%5E4+%28x%29

sin%5E2+%28x%29%281+%2B+%28+1+-+sin%5E2+%28x%29%29%29 --- Replacing  

sin%5E2+%28x%29%282+-+sin%5E2+%28x%29%29

highlight_green%282sin%5E2+%28x%29+-+sin%5E4+%28x%29%29 ------- DISTRIBUTING (Same as RHS)