Question

Prove that :
$(\sin \theta +\cos \theta )(\tan \theta +\cot \theta )=\sec \theta +\csc \theta$.

Answer 1

To prove:- $(\sin \theta +\cos \theta )(\tan \theta +\cot \theta )=\sec \theta +\csc \theta$

Proof:-

Taking L.H.S., we have

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

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

$=(\sin \theta +\cos \theta )\left(\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta }\right)$

$=(\sin \theta +\cos \theta )\left(\frac{1}{\sin \theta \cos \theta }\right)(\because {\sin }^2\theta +{\cos }^2\theta =1)$

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

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

$=\sec \theta +\csc \theta$

$=$ R.H.S.

Hence proved.