Answer
$$2\cos^2\frac{x}{2}\tan x=\tan x+\sin x$$
2 sides are equal as proved below, so the equation is an identity.
Work Step by Step
$$2\cos^2\frac{x}{2}\tan x=\tan x+\sin x$$
Again, we examine from the left side.
$$X=2\cos^2\frac{x}{2}\tan x$$
- For $\cos^2\frac{x}{2}$:
From the half-angle identity for cosine: $$\cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}}$$
Thus, $$\cos^2\frac{x}{2}=\frac{1+\cos x}{2}$$ (After doubling, the sign is always positive)
- For $\tan x$:
From the quotient identity: $$\tan x=\frac{\sin x}{\cos x}$$
Apply them back to $X$:
$$X=2\times\frac{1+\cos x}{2}\times\frac{\sin x}{\cos x}$$
$$X=\frac{(1+\cos x)\sin x}{\cos x}$$
$$X=\frac{\sin x+\sin x\cos x}{\cos x}$$
Now we separate the fraction.
$$X=\frac{\sin x}{\cos x}+\frac{\sin x\cos x}{\cos x}$$
$$X=\tan x+\sin x$$ (as $\tan x=\frac{\sin x}{\cos x}$)
Therefore, $$2\cos^2\frac{x}{2}\tan x=\tan x+\sin x$$
2 sides are equal, so the equation is verified to be an identity.
