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Prove the Following Trigonometric Identities (1 + Tan^2 Theta)/(1 + Cot^2 Theta) = ((1 - Tan Theta)/(1 - Cot Theta))^2 = Tan^2 Theta - CBSE Class 10 - Mathematics

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Question

Prove the following trigonometric identities

`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`

Solution

We have to prove `(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`

Consider the expression

`(1 + tan^2 theta)/(1 + cot^2 theta) = (1 + tan^2 theta)/(1 + 1/(tan^2 theta))`

= `(1 +tan^2 theta)/((tan^2 theta + 1)/tan^2 theta)`

`= tan^2 theta (1 + tan^2 theta)/(1 + tan^2 theta)`

`= tan^2 theta`

Again, we have

`((1 - tan theta)/(1 - cot theta))^2 = ((1 - tan theta)/(1 - 1/(tan theta)))^2`

`= ((1 - tan theta)/((tan theta - 1)/tan theta))`

`= tan^2 theta ((1 - tan theta)/(tan theta - 1))^2`

`= tan^2 theta ((1 - tan theta)/(1 -  tan theta))^2`

`= tan^2 theta(-1)^2`

`= tan^2 theta`

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Solution Prove the Following Trigonometric Identities (1 + Tan^2 Theta)/(1 + Cot^2 Theta) = ((1 - Tan Theta)/(1 - Cot Theta))^2 = Tan^2 Theta Concept: Trigonometric Identities.
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