Prove the following trigonometric identities. cos^2A+1 1+ ^2A =1 - Vidyadip

Question

Prove the following trigonometric identities.
$\text{cos}^2\text{A}+\frac{1}{1+\cot^2\text{A}}=1$

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Answer

We know that,
$\sin^2\text{A}+\cos^2\text{A}=1,$
$\text{cosec}^2\text{A}-\text{cot}^2\text{A}=1$
So,
$\text{L.H.S.} = \text{cos}^2\text{A}+\frac{1}{1+\cot^2\text{A}}$
$\cos^2\text{A}+\frac{1}{1+\cot^2\text{A}}=\cos^2\text{A}+\frac{1}{\text{cosec}^2\text{A}}$
$=\cos^2\text{A}+\Big(\frac{1}{\text{cosec A}}\Big)^2$
$=\cos^2\text{A}+(\sin\text{ A})^2$
$=\cos^2\text{A}+\sin^2\text{A}$
$=1 = \text{R.H.S.}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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