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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Updated on: 10-Oct-2022

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