Question

Differentiating the identity $\sin 2x=2\sin x\cos x$ term-by term, prove the identity $\cos 2x$ = ${\cos }^{2}x$ - ${\sin }^{2}x$ .

Answer 1

$\sin 2x=2\sin x\cos x$

Aim : - To differentiate the identity to prove

$\cos 2x={\cos }^{2}x-{\sin }^{2}x$

$\sin 2x=2\sin x\cos x$

differentiate both sides with respect to $x$

$2\cos 2x=2\sin x(-\sin x)+2\cos x\cos x$

$2\cos 2x=2[{\cos }^{2}x-{\sin }^{2}x]$

$\cos 2x={\cos }^{2}x-{\sin }^{2}x$

Hence proved.