(i) \(\because\) sin(x + y) = sin x cos y + cos x sin y
put y = x, we obtain
sin 2x = sin x cos x + cos x sin x
= 2sin x cos x
(ii) \(\because\) sin 2x = 2sin x cos x
= 2 \(\frac{\sin \mathrm x}{\cos \mathrm x}\) cos2x
= \(\frac{2\tan\mathrm x}{\sec^2\mathrm x}\) (\(\because\) sec x = \(\frac{1}{\cos\mathrm x}\) & tan x = \(\frac{\sin\mathrm x}{\cos\mathrm x}\))
\(=\frac{2\tan\mathrm x}{1+\tan^2\mathrm x}\) (\(\because\) sec2x = 1 + tan2x)