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