Prove the following trigonometric identities:$ \tan \theta+\frac{1}{\tan \theta}=\sec \theta \operatorname{cosec} \theta $

To do:

We have to prove that \( \tan \theta+\frac{1}{\tan \theta}=\sec \theta \operatorname{cosec} \theta \).

Solution: We know that,

$\sec ^{2} A-tan ^{2} A=1$.......(i)

$ \tan A=\frac{\sin\ A}{\cos\ A}=\frac{\sec\ A}{\operatorname{cosec} A}$.......(ii)

Therefore,

$\tan \theta+\frac{1}{\tan \theta}=\frac{\tan ^{2} \theta+1}{\tan \theta}$

$=\frac{\sec ^{2} \theta}{\tan \theta}$      (From (i))

$=\sec \theta \times \frac{\sec \theta}{\sec \theta} \times \operatorname{cosec} \theta$           (From (ii))

$=\sec \theta \operatorname{cosec} \theta$             

Hence proved.  

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

82 Views

Kickstart Your Career

Get certified by completing the course

Get Started