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