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.