## Trigonometry (11th Edition) Clone

$$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$ The equation is an identity, as proved below.
$$\tan\frac{\theta}{2}=\csc\theta-\cot\theta$$ 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.