Question

Evaluate:
${\int }_0^{\pi /4}({sin}^2\frac{x}{2}-co{s}^2\frac{x}{2})dx$

Answer 1

We know that ${\cos }^{2}x-{\sin }^{2}x=\cos 2x$

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

${\int }_0^{\frac{\pi }{4}}-({\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2})dx$

$=-{\int }_0^{\frac{\pi }{4}}\cos xdx$

$=-{\left[\sin x\right]}_0^{\frac{\pi }{4}}$

$=-[\sin \frac{\pi }{4}-\sin 0]$

$=-[\frac{1}{\sqrt{2}}-0]$

$=\frac{-1}{\sqrt{2}}$