Prove that

To Prove: $2 \sin 22 \frac{1^{\circ}}{2} \cos 22 \frac{1^{\circ}}{2}=\frac{1}{\sqrt{2}}$

Taking LHS,

$=2 \sin 22 \frac{1}{2}^{\circ} \cos 22 \frac{1}{2}^{\circ} \ldots$ (i)

We know that,

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

Here, $\mathrm{x}=22 \frac{1}{2}=\frac{45}{2}$

So, eq. (i) become

$=\sin 2\left(\frac{45}{2}\right)$

$=\sin 45^{\circ}$

$=\frac{1}{\sqrt{2}}\left[\because \sin \left(45^{\circ}\right)=\frac{1}{\sqrt{2}}\right]$

= RHS

∴ LHS = RHS

Hence Proved