Question

Prove the identity
$(\sin \theta +\cos \theta )(\tan \theta +\cot \theta )=\sec \theta +\csc \theta$

Answer 1

$(\sin \theta +\cos \theta )(\tan \theta +\cot \theta )$

$(\sin \theta +\cos \theta )(\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta })$

$(\sin \theta +\cos \theta )\left(\frac{1}{\sin \theta \cos \theta }\right)$

$\frac{\sin \theta +\cos \theta }{\sin \theta \cos \theta }$

$\frac{\sin \theta }{\sin \theta \cos \theta }+\frac{\cos \theta }{\sin \theta \cos \theta }$

$\frac{1}{\cos \theta }+\frac{1}{\sin \theta }=\sec \theta +cosec\theta$

$=\sec \theta +cosec\theta$.