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Question

$$\frac{sin 2 x}{1+cos 2x}=tan x$$


Solution

$$\cfrac{\sin{2x}}{1 + \cos{2x}} = \tan{x}$$
Taking L.H.S.
$$\cfrac{\sin{2x}}{1 + \cos{2x}}$$
Using trigonometric identity:
$$\sin{2x} = 2 \sin{x} \cos{x}$$
$$\cos{2x} = 2 \cos^{2}{x} - 1 \Rightarrow 1 + \cos{2x} = 2 \cos^{2}{x}$$
Therefore,
$$\cfrac{\sin{2x}}{1 + \cos{2x}}$$
$$= \cfrac{2 \sin{x} \cos{x}}{2 \cos^{2}{x}}$$
$$= \cfrac{\sin{x}}{\cos{x}}$$
$$= \tan{x}$$
$$= $$R.H.S.
Hence proved that $$\cfrac{\sin{2x}}{1 + \cos{2x}} = \tan{x}$$.

Mathematics

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