Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Prove the Following Identities 1 − Sin 2 X 1 + Cot X − Cos 2 X 1 + Tan X = Sin X Cos X - Mathematics

Prove the following identities
$1 - \frac{\sin^2 x}{1 + \cot x} - \frac{\cos^2 x}{1 + \tan x} = \sin x \cos x$

#### Solution

$1 - \frac{\sin^2 x}{1 + \cot x} - \frac{\cos^2 x}{1 + \tan x} = \sin x \cos x$
$\text{ LHS }= 1 - \frac{\sin^3 x}{\sin x + \cos x} - \frac{\cos^3 x}{\sin x + \cos x}$
$= \frac{\sin x + \cos x - \left( \sin^3 x + \cos^3 x \right)}{\sin x + \cos x}$
$= \frac{\left( \sin x + \cos x \right)\left( 1 - \sin^2 x - \cos^2 x + \sin x \cos x \right)}{\sin x + \cos x}$
$= \left( 1 - \sin^2 x - \cos^2 x + \sin x \cos x \right)$
$= \left( 1 - 1 + \sin x \cos x \right)$
$= \sin x \cos x$
= RHS
Hence proved.

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#### APPEARS IN

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