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Q.

State true or false.
the following trigonometric expression:

$\frac{{cosec}^{2} θ - \cos^{2} θ}{\cot^{2} θ} = \sec^{2} θ - \sin^{2} θ$

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a

True

b

False

answer is A.

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Detailed Solution

Given expression is,

\frac{{cosec}^{2} θ - \cos^{2} θ}{\cot^{2} θ} = \sec^{2} θ - \sin^{2} θ

Take the left hand side,

\frac{{cosec}^{2} θ - \cos^{2} θ}{\cot^{2} θ}

= \frac{{cosec}^{2} θ}{\cot^{2} θ} - \frac{\cos^{2} θ}{\cot^{2} θ}

= \frac{\frac{1}{\sin^{2} θ}}{\frac{\cos^{2} θ}{\sin^{2} θ}} - \frac{\cos^{2} θ}{\frac{\cos^{2} θ}{\sin^{2} θ}}

(

cosec θ = \frac{1}{\sin θ}, \cot θ = \cos θ / \sin θ

)

= \sec^{2} θ - \sin^{2} θ

Which is equal to right hand side. So, the given relation is true.
Hence, Proved Correct option is 1.

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