If A = 45°, verify that:

$A=45^{\circ}$

$\Rightarrow 2 A=2 \times 45^{\circ}=90^{\circ}$

(i) $\sin 2 A=\sin 90^{\circ}=1$

$2 \sin A \cos A=2 \sin 45^{\circ} \cos 45^{\circ}=2 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}=1$

∴ sin 2A = 2 sin A cos A

(ii) cos 2A = cos 90o = 0

$2 \cos ^{2} A-1=2 \cos ^{2} 45^{\circ}-1=2 \times\left(\frac{1}{\sqrt{2}}\right)^{2}-1=2 \times \frac{1}{2}-1=1-1=0$

Now, $1-2 \sin ^{2} A=1-2 \times\left(\frac{1}{\sqrt{2}}\right)^{2}=1-2 \times \frac{1}{2}=1-1=0$

$\therefore \cos 2 A=2 \cos ^{2} A-1=1-2 \sin ^{2} A$