# Prove the Following Trigonometric Identities. (Tan^2 A)/(1 + Tan^2 A) + (Cot^2 A)/(1 + Cot^2 A) = 1 - Mathematics

Prove the following trigonometric identities.

(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1

#### Solution

In the given question, we need to prove (tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1

Here, we will first solve the LHS.

Now using tan theta = sin theta/cos theta and cot theta = cos theta/sin theta we get

tan^2 A/(1 + tan^2 A) + cot^2 A/(1 + cot^2 A) = ((sin^2 A/cos^2 A))/((1 + sin^2 A/cos^2 A)) + ((cos^2 A/sin^2 A))/((1 + cos^2 A/sin^2 A))

= ((sin^2 A/cos^2 A))/(((cos^2 + sin^2 A)/cos^2 A)) + ((cos^2 A/sin^2 A))/(((sin^2 A + cos^2 A)/sin^2 A))

= ((sin^2 A/cos^2 A))/((1/cos^2 A)) + ((cos^2 A/sin^2 A))/((1/(sin^2  A)))    (using sin^2 theta + cos^2 theta = 1)

On further solving by taking the reciprocal of the denominator, we get,

(sin^2 A/cos^2 A)/(1/cos^2 A) + (cos^2 A/sin^2 A)/(1/sin^2 A) = ((sin^2 A)/(cos^2 A)) (cos^2 A/1) + (cos^2 A/sin^2 A)(sin^2 A/1)

= sin^2 A + cos^2 A        (Using sin^2 theta + cos^2 theta = 1)

= 1

Hence proved.

Concept: Trigonometric Identities
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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 11 Trigonometric Identities
Exercise 11.1 | Q 45 | Page 45