Question

Prove the identity:
$(\sin \theta +\cos \theta )(\tan \theta +\cot \theta )=\sec \theta +\text{cosec}\theta$

Answer 1

L.H.S. $=(\sin \theta +\cos \theta )(\tan \theta +\cot \theta )$
$=(\sin \theta +\cos \theta )(\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta })$
$=(\sin \theta +\cos \theta )\left(\frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta \sin \theta }\right)$
$=(\sin \theta +\cos \theta )\times \frac{1}{\cos \theta \sin \theta }$
$=\frac{\sin \theta }{\cos \theta \sin \theta }+\frac{\cos \theta }{\cos \theta \sin \theta }$
$=\frac{1}{\cos \theta }+\frac{1}{\sin \theta }=\sec \theta +\text{cosec}\theta =$ R.H.S