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Prove the following trigonometric identities:$ \cos ^{2} A+\frac{1}{1+\cot ^{2} A}=1 $
To do:
We have to prove that \( \cos ^{2} A+\frac{1}{1+\cot ^{2} A}=1 \).
Solution:
We know that,
$\operatorname{cosec} ^{2} A-cot ^{2} A=1$.......(i)
$ \sin A=\frac{1}{\operatorname{cosec} A}$.......(ii)
$\sin ^{2} A+\cos ^{2} A=1$.......(iii)
Therefore,
$\cos ^{2} A+\frac{1}{1+\cot ^{2} A}=\cos ^{2} A+\frac{1}{\operatorname{cosec} ^{2} A}$ (From (i))
$=\cos ^{2} A+\sin ^{2} A$ (From (ii))
$=1$ (From (iii))
Hence proved.
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