Question

Prove that $\tt \: \dfrac{ {cot}^{2}A }{cosec A - 1} - 1 = cosec A$​

Answer 1

To prove:-

• $\sf \: \dfrac{ {cot}^{2}A }{cosec A - 1} - 1 = cosec A$

Formula Used:-

• $\boxed{ \purple{ \bf \: {cot}^{2} = {cosec }^{2} A - 1}}$
• $\boxed{ \purple{ \bf \: {x}^{2} - {y}^{2} = (x + y)(x - y)}}$

Proof:-

$\sf:\implies LHS$

$\sf:\implies\dfrac{ {cot}^{2}A }{cosec A - 1} - 1$

$\sf:\implies\dfrac{ {cosec }^{2} A - 1}{cosec A - 1} - 1$

$\sf:\implies\dfrac{(cosec A + 1) \: \: \cancel{(cosec A - 1)}}{ \cancel{cosec A - 1}} - 1$

$\sf:\implies cosec A \: + \: \cancel 1 \: - \: \cancel1$

$\sf:\implies cosec A$

$\sf:\implies RHS$

• Hence,Proved

___________________________________________________