Question
Prove the following identities:
$\sin^2\theta+\cos^4\theta=\cos^2\theta+\sin^4\theta$
$\sin^2\theta+\cos^4\theta=\cos^2\theta+\sin^4\theta$
✓
Answer
$\text{LHS}=\sin^2\theta+\cos^4\theta$
$=\sin^2\theta+\big(\cos^2\theta\big)^2$
$=\sin^2\theta+\big(1-\sin^2\theta\big)^2$
$=\sin^2\theta+1-2\sin^2\theta+\sin^4\theta$
$=1-\sin^2\theta+\sin^4\theta$
$=\cos^2\theta+\sin^4\theta$
$=\text{R.H.S}$
Hence, $\text{R.H.S.}=\text{L.H.S.}$
$=\sin^2\theta+\big(\cos^2\theta\big)^2$
$=\sin^2\theta+\big(1-\sin^2\theta\big)^2$
$=\sin^2\theta+1-2\sin^2\theta+\sin^4\theta$
$=1-\sin^2\theta+\sin^4\theta$
$=\cos^2\theta+\sin^4\theta$
$=\text{R.H.S}$
Hence, $\text{R.H.S.}=\text{L.H.S.}$
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