Question

Prove the following identity:

Answer 1

Answer:

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Answer 2

ANSWER:-

Given:

If x= asin∅, y= btan∅ .

To prove:

$\frac{ {a}^{2} }{ {x}^{2} } - \frac{ {b}^{2} }{ {y}^{2} } = 1$

Proof:

Take L.H.S

$\frac{ {a}^{2} }{ {x}^{2} } - \frac{ {b}^{2} }{ {y}^{2} } \\ \\ = > \frac{ {a}^{2} }{ {a}^{2} {sin}^{2} \theta } - \frac{ {b}^{2} }{ {b}^{2} {tan}^{2} \theta} \\ \\ = > \frac{1}{ {sin}^{2} \theta } - \frac{1}{ {tan}^{2} \theta } \\ \\ = > \frac{ {tan}^{2} \theta - {sin}^{2} \theta }{ {sin}^{2} \theta \: { tan }^{2} \theta} \\ \\ = > \frac{ {tan}^{2} \theta(1 - \frac{ {sin}^{2} \theta }{ {tan}^{2} \theta } ) }{ {tan}^{2} \theta {sin}^{2} \theta } \\ \\ = > \frac{1 - {cos}^{2} \theta}{ {sin}^{2} \theta} \\ \\ = > \frac{ {sin}^{2} \theta}{ {sin}^{2 } \theta } \\ \\ = > 1 \: \: \: \: \: \: [R.H.S]$

Hence,

Proved.

Hope it helps ☺️