Answer
$$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$
The equation is an identity, as proved below.
Work Step by Step
$$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$
We examine the right side first.
$$X=\csc\theta-\cot\theta$$
- Reciprocal Identity: $\csc\theta=\frac{1}{\sin\theta}$
- Quotient Identity: $\cot\theta=\frac{\cos\theta}{\sin\theta}$
Apply the identities to $X$:
$$X=\frac{1}{\sin\theta}-\frac{\cos\theta}{\sin\theta}$$
$$X=\frac{1-\cos\theta}{\sin\theta}$$
- Half-angle Identity for tangent: $\frac{1-\cos\theta}{\sin\theta}=\tan\frac{\theta}{2}$
Therefore, $$X=\tan\frac{\theta}{2}$$
So 2 sides are equal.
$$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$
The equation is an identity.
