Prove the following.
prove that cot2θ – tan2θ = cosec2θ – sec2θ

Chapter 6 – Trigonometry – Text Book Solution

Problem set 6| Q 5.4| Page 138

Prove the following.
(4) prove that cot²θ – tan²θ = cosec²θ – sec²θ

Solution

[cot^2 theta – tan^2 theta]

[ = left( {cosec}^2 theta – 1 right) – left( sec^2 theta – 1 right) left( 1 + tan^2 theta = sec^2 theta   & 1 + cot^2 theta = {cosec}^2 theta right)]

[ = {cosec}^2 theta – 1 – sec^2 theta + 1]

[ = {cosec}^2 theta – sec^2 theta]

Solution

The expression we are given is cot^2θ – tan^2θ. We can use trigonometric identities to simplify this expression.

We know that 1 + tan^2θ = sec^2θ and 1 + cot^2θ = cosec^2θ. Using these identities, we can rewrite the expression as follows:

cot^2θ – tan^2θ = (1 + cot^2θ) – (1 + tan^2θ) [Using the identities above] = (cosec^2θ – 1) – (sec^2θ – 1) [Rewriting in terms of cosecant and secant] = cosec^2θ – sec^2θ [Simplifying]

Therefore, we have shown that cot^2θ – tan^2θ is equal to cosec^2θ – sec^2θ.

Chapter 6 – Trigonometry – Text Book Solution

Problem Set 6 |Q 5.4| P 138

Q.5.3


Q 5.4


Q.5.5

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