Prove the following trigonometric identities:$ \left(1+\cot ^{2} A\right) \sin ^{2} A=1 $

To do:

We have to prove that \( \left(1+\cot ^{2} A\right) \sin ^{2} A=1 \).

Solution:
We know that,

$ \operatorname{cosec}^{2} A-\cot ^{2} A=1$.......(i)

$\sin ^{2} A\times\operatorname{cosec}^{2} A=1$.......(ii)

Therefore,

$\left(1+\cot ^{2} A\right) sin ^{2} A=(\operatorname{cosec}^{2} A)(\sin ^{2} A)$       (From (i))

$=\sin ^{2} A\times\operatorname{cosec}^{2} A$              (From (ii))

$=1$

Hence proved.