Prove that: ^2( 4 -x)- ^2( 4 -x)= 2x - Vidyadip

Question

Prove that:
$\cos^2(\frac{\pi}{4}-\text{x})-\sin^2(\frac{\pi}{4}-\text{x})=\sin2\text{x}$

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Answer

$\text{LHS}=\cos^2\Big(\frac{\pi}{4}-\text{x}\Big)-\sin^2\Big(\frac{\pi}{4}-\text{x}\Big)$
$=\cos2\Big(\frac{\pi}{4}-\text{x}\Big)$ $\Big[\because\cos2\text{x}=\cos^2\theta-\sin^2\text{x}\Big]$
$=\cos(\frac{\pi}{2}-2\text{x})$ $\Big[\because\cos\Big(\frac{\pi}{2}-\text{x}\Big)=\sin\text{x}\Big]$
$=\sin2\text{x}=\text{RHS}$

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