Prove the following trigonometric identities. ^2A+ ^2A= ^2A cosec… - Vidyadip

Question

Prove the following trigonometric identities.
$\tan^2\text{A}+\cot^2\text{A}=\sec^2\text{A}\text{ cosec}^2\text{A}-2$

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Answer

$\text{L.H.S}=\tan^2\text{A}+\cot^2\text{A}$
$=\sec^2\text{A}-1+\text{cosec}^2\text{A}-1$
$=\sec^2\text{A}+\text{cosec}^2\text{A}-2$
$=\frac{1}{\cos^2\text{A}}+\frac{1}{\sin^2\text{A}}-2$
$=\frac{\sin^2\text{A}+\cos^2\text{A}}{\cos^2\text{A}\sin^2\text{A}}-2$
$=\frac{1}{\cos^2\text{A}\sin^2\text{A}}-2$
$=\sec^2\text{A}\text{ cosec}^2\text{A}-2=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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