Question

If A = 45°, verify that:
(i) sin 2A = 2 sin A cos A
(ii) cos 2A = 2 cos² A − 1 = 1 − 2 sin² A

Answer 1

A = 45^o
⇒ 2A = 2 $\times$ 45^o = 90^o

(i) sin 2A = sin 90^o = 1
2 sin A cos A = 2 sin 45^o cos 45^o = $2\times \frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}=1$
∴ sin 2A = 2 sin A cos A

(ii) cos 2A = cos 90^o = 0
2 cos² A − 1 = 2 cos² 45^o − 1 = $2\times {\left(\frac{1}{\sqrt{2}}\right)}^2-1=2\times \frac{1}{2}-1=1-1=0$
Now, 1 − 2 sin² A = $1-2\times {\left(\frac{1}{\sqrt{2}}\right)}^2=1-2\times \frac{1}{2}=1-1=0$
∴ cos 2A = 2 cos² A − 1 = 1 − 2 sin² A