Question

Show that sin^4theta - cos^4 theta = 1 - 2cos^2 theta.

Answer 1

Given :

$sin^4\theta-cos^4\theta$

$\implies sin^4\theta+cos^4\theta+2sin^2\theta cos^2\theta-2sin^2\theta cos^2\theta$

[ We know that : $a^2+b^2+2ab=(a+b)^2$ ]

$\implies (sin^2\theta+cos^2\theta)^2-2sin^2\theta cos^2\theta$

$\implies 1-2sin^2\theta cos^2\theta$

Hence Proved !

Hope it helps :-)

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Answer 2

let

sin^2 theta = K
cos^2 theta = U
then,

Taking L.H.S,

sin^4 theta - cos^4 theta
K^2 - U^2
(K+U) (K-U)
(sin^2 theta + cos^2 theta) (sin^2 theta- cos^2 theta)
( 1 ) (1-cos^2 theta - cos^2 theta)

1- 2 cos^2 theta

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