Question

prove that cos 4 theta minus sin 4 theta equals to 1 - 2 sin squared theta

Answer 1

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Answer 2

$\cos ^{4} \theta-\sin ^{4} \theta=1-2\sin ^{2} \theta$.

Step-by-step explanation:

L.H.S.= $\cos ^{4} \theta-\sin ^{4} \theta$

$=( \cos ^{2})^{2} \theta-(\sin ^{2} \theta)^{2}$

$=(\cos ^{2} \theta+\sin ^{2} \theta)(\cos ^{2}  \theta-\sin ^{2} \theta)$

[ ∵ $a^{2} -b^{2} =(a+b)(a-b)$]

$=(1)(\cos ^{2} \theta-\sin ^{2} \theta)$

[ ∵ $\cos ^{2} \theta+\sin ^{2} \theta=1$]

$=(1-\sin ^{2} \theta-\sin ^{2} \theta)$

$=1-2\sin ^{2} \theta$

= R.H.S., proved.

Hence,$\cos ^{4} \theta-\sin ^{4} \theta=1-2\sin ^{2} \theta$.