Question

Prove the following:
$\left(\sin \alpha +\cos \alpha \right)\left(\tan \alpha +\cot \alpha \right)=\sec \alpha +\cos ec\alpha$

Answer 1

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