Prove the Following Identities C O S E C X ( Sec X − 1 ) − Cot X ( 1 − Cos X ) = Tan X − Sin X - Mathematics

Prove the following identities
$cosec x \left( \sec x - 1 \right) - \cot x \left( 1 - \cos x \right) = \tan x - \sin x$

Solution

$\text{ LHS }= cosec x \left( \sec x - 1 \right) - \cot x \left( 1 - \cos x \right)$
$= \frac{1}{\sin x} \left( \frac{1}{\cos x} - 1 \right) - \frac{\cos x}{\sin x} \left( 1 - \cos x \right)$
$= \frac{1}{\sin x} \left( \frac{1 - \cos x}{\cos x} \right) - \frac{\cos x}{\sin x} \left( 1 - \cos x \right)$
$= \left( \frac{1 - \cos x}{\sin x} \right)\left( \frac{1}{\cos x} - \cos x \right)$
$= \left( \frac{1 - \cos x}{\sin x} \right)\left( \frac{1 - \cos^2 x}{\cos x} \right)$
$= \left( \frac{1 - \cos x}{\sin x} \right)\left( \frac{\sin^2 x}{\cos x} \right)$
$= \left( 1 - \cos x \right)\left( \frac{\sin x}{\cos x} \right)$
$= \frac{\sin x}{\cos x} - \sin x$
$= \tan x - \sin x$
= RHS
Hence proved.

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 5 Trigonometric Functions
Exercise 5.1 | Q 4 | Page 18