Prove the Following Identities ( 1 + Tan α Tan β ) 2 + ( Tan α − Tan β ) 2 = Sec 2 α Sec 2 β - Mathematics

Question

Prove the following identities
\[\left( 1 + \tan \alpha \tan \beta \right)^2 + \left( \tan \alpha - \tan \beta \right)^2 = \sec^2 \alpha \sec^2 \beta\]

Solution

\[\left( 1 + \tan \alpha \tan \beta \right)^2 + \left( \tan \alpha - \tan \beta \right)^2 = \sec^2 \alpha \sec^2 \beta\]

\[\text{ LHS }= \left( 1 + \tan\alpha \tan\beta \right)^2 + \left( \tan\alpha - \tan\beta \right)^2 \]

\[ = 1 + \tan^2 \alpha \tan^2 \beta + 2\tan\alpha \tan\beta + \tan^2 \alpha + \tan^2 \beta - 2\tan\alpha \tan\beta\]

\[ = 1 + \tan^2 \alpha \tan^2 \beta + \tan^2 \alpha + \tan^2 \beta\]

\[ = \tan^2 \alpha\left( \tan^2 \beta + 1 \right) + 1\left( 1 + \tan^2 \beta \right)\]

\[ = \left( 1 + \tan^2 \beta \right)\left( 1 + \tan^2 \alpha \right)\]

\[ = \sec^2 \alpha . \sec^2 \beta \]

= RHS

Hence proved.

shaalaa.com

Trigonometric Functions - Truth of the Identity

Is there an error in this question or solution?

Q 13 Q 12 Q 14

Chapter 5: Trigonometric Functions - Exercise 5.1 [Page 18]

APPEARS IN

RD Sharma Mathematics [English] Class 11

Chapter 5 Trigonometric Functions
Exercise 5.1 | Q 13 | Page 18

RELATED QUESTIONS

Prove the following identites

sec⁴x - sec²x = tan⁴x + tan²x

Prove the following identities
\[\sin^6 x + \cos^6 x = 1 - 3 \sin^2 x \cos^2 x\]

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

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

Prove the following identities
\[\frac{1 - \sin x \cos x}{\cos x \left( \sec x - cosec x \right)} \cdot \frac{\sin^2 x - \cos^2 x}{\sin^3 x + \cos^3 x} = \sin x\]

Prove the following identitie

\[\frac{\tan x}{1 - \cot x} + \frac{\cot x}{1 - \tan x} = \left( \sec x cossec x + 1 \right)\]

Prove the following identities
\[\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} + \frac{\sin^3 x - \cos^3 x}{\sin x - \cos x} = 2\]

Prove the following identities
\[\left( \sec x \sec y + \tan x \tan y \right)^2 - \left( \sec x \tan y + \tan x \sec y \right)^2 = 1\]

Prove the following identities
\[\frac{\cos x}{1 - \sin x} = \frac{1 + \cos x + \sin x}{1 + \cos x - \sin x}\]