![Trending Articles on Technical and Non Technical topics](/images/trending_categories.jpeg)
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 the following:
$ 1+\frac{\cot ^{2} \alpha}{1+\operatorname{cosec} \alpha}=\operatorname{cosec} \alpha $
To do:
We have to prove that \( 1+\frac{\cot ^{2} \alpha}{1+\operatorname{cosec} \alpha}=\operatorname{cosec} \alpha \).
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,
$1+\frac{\cot ^{2} \alpha}{1+\operatorname{cosec} \alpha}=1+\frac{\frac{\cos ^{2} \alpha}{\sin ^{2} \alpha}}{1+\frac{1}{\sin \alpha}}$
$=1+\frac{\frac{\cos ^{2} \alpha}{\sin ^{2} \alpha}}{\frac{\sin \alpha+1}{\sin \alpha}}$
$=1+\frac{\cos ^{2} \alpha}{\sin ^{2} \alpha} \times \frac{\sin \alpha}{\sin \alpha+1}$
$=1+\frac{\cos ^{2} \alpha}{\sin \alpha(\sin \alpha+1)}$
$=\frac{\sin \alpha(\sin \alpha+1)+\cos ^{2} \alpha}{\sin \alpha(\sin \alpha+1)}$
$=\frac{\sin^2 \alpha+\sin \alpha+\cos ^{2} \alpha}{\sin \alpha(\sin \alpha+1)}$
$=\frac{1+\sin \alpha}{\sin \alpha(\sin \alpha+1)}$
$=\frac{1}{\sin \alpha}$
$=\operatorname{cosec} \alpha$
Hence proved.