Question

Prove that tan theta /1+costheta+ sin theta/1-cos theta=cottheta+cosectheta*sectheta

Answer 1

Here

$LHS=\frac{\tan \theta}{1+\cos \theta} +\frac{\sin \theta}{1-\cos \theta} \\ LHS=\frac{\tan \theta(1-\cos \theta)+\sin \theta (1+\cos \theta)}{(1+\cos \theta)(1-\cos \theta)} \\ LHS=\sin \theta \frac{\sec \theta(1-\cos \theta)+(1+\cos \theta)}{1-\cos^2 \theta} \\ LHS=\sin \theta \frac{\sec \theta-1+(1+\cos \theta)}{\sin^2 \theta} \\ LHS=\frac{\sec \theta+\cos \theta}{\sin \theta} \\ LHS=\frac{\sec \theta}{\sin \theta} +\frac{\cos \theta}{\sin \theta} \\ LHS=\cot \theta+\csc \theta \sec \theta=RHS$

The proof is complete.