Prove the following: (90^ -A) cosec^2A - ^2A=0 - Vidyadip

Question

Prove the following:
$\frac{\tan(90^\circ-\text{A})\cot\text{A}}{\text{cosec}^2\text{A}}-\cos^2\text{A}=0$

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Answer

We have to prove: $\frac{\tan(90^\circ-\text{A})\cot\text{A}}{\text{cosec}^2\text{A}}-\cos^2\text{A}=0$
Left hand side
$=\frac{\tan(90^\circ-\text{A})\cot\text{A}}{\text{cosec}^2\text{A}}-\cos^2\text{A}$
$=\frac{\cot\text{A}.\cot\text{A}}{\text{cosec}^2\text{A}}-\cos^2\text{A}$
$=\frac{\cot^2\text{A}}{\text{cosec}^2\text{A}}-\cos^2\text{A}$
$=\frac{\cos^2\text{A}.\sin^2\text{A}}{\sin^2\text{A}}-\cos^2\text{A}$
$=\cos^2\text{A}-\cos^2\text{A}$
$=0$
= Right hand side
Proved.

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