Question

Differentiate the function given below w.r.t. $x:$

$(\sin x+\cos x{)}^2$

Answer 1

Let $y=(\sin x+\cos x{)}^2$

Differentiating with respect to $x,$ we get

$\frac{dy}{dx}=\frac{d}{dx}(\sin x+\cos x{)}^2$

$=2(\cos x+\sin x)\frac{d}{dx}(\cos x+\sin x)$

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

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

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