- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Prove that:
\( \frac{\cos \theta}{\operatorname{cosec} \theta+1}+\frac{\cos \theta}{\operatorname{cosec} \theta-1}=2 \tan \theta \)
To do:
We have to prove that \( \frac{\cos \theta}{\operatorname{cosec} \theta+1}+\frac{\cos \theta}{\operatorname{cosec} \theta-1}=2 \tan \theta \).
Solution:
We know that,
$\sin^2 A+\cos^2 A=1$
$\operatorname{cosec}^2 A-\cot^2 A=1$
$\sec^2 A-\tan^2 A=1$
$\cot A=\frac{\cos A}{\sin A}$
$\tan A=\frac{\sin A}{\cos A}$
$\operatorname{cosec} A=\frac{1}{\sin A}$
$\sec A=\frac{1}{\cos A}$
Therefore,
$\frac{\cos \theta}{\operatorname{cosec} \theta+1}+\frac{\cos \theta}{\operatorname{cosec} \theta-1}=\frac{\cos \theta}{\frac{1}{\sin \theta}+1}+\frac{\cos \theta}{\frac{1}{\sin \theta}-1}$
$=\frac{\cos \theta}{\frac{1+\sin \theta}{\sin \theta}}+\frac{\cos \theta}{\frac{1-\sin \theta}{\sin \theta}}$
$=\frac{\sin \theta \cos \theta}{1+\sin \theta}+\frac{\sin \theta \cos \theta}{1-\sin \theta}$
$=\sin \theta \cos \theta\left[\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}\right]$
$=\sin \theta \cos \theta\left[\frac{1-\sin \theta+1+\sin \theta}{(1+\sin \theta)(1-\sin \theta)}\right]$
$=\sin \theta \cos \theta\left[\frac{2}{1-\sin ^{2} \theta}\right]$
$=\frac{2\sin \theta \cos \theta}{\cos ^{2} \theta}$
$=\frac{2 \sin \theta}{\cos \theta}$
$=2 \tan \theta$
Hence proved.