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