Answer
$$\tan\frac{x}{2}=\csc x-\cot x$$
As proved in the Work Step by Step, the equation is an identity.
Work Step by Step
$$\tan\frac{x}{2}=\csc x-\cot x$$
The right side would be examined first here.
$$X=\csc x-\cot x$$
We can rewrite both $\csc x$ and $\cot x$ according to the following identities:
$$\csc x=\frac{1}{\sin x}\hspace{2cm}\cot x=\frac{\cos x}{\sin x}$$
Therefore,
$$X=\frac{1}{\sin x}-\frac{\cos x}{\sin x}$$
$$X=\frac{1-\cos x}{\sin x}$$
Now recall 3 half-angle identities for tangent, one of which is $$\tan\frac{x}{2}=\frac{1-\cos x}{\sin x}$$
Therefore, $$X=\tan\frac{x}{2}$$
Hence, $$\tan\frac{x}{2}=\csc x-\cot x$$
The equation is an identity, verified by the fact 2 sides of the equation are proved to be equal.
