Question : Simplify the given equation: $(1+\tan ^2 A)(1+\cot ^2 A)=?$
Option 1: $\frac{1}{\cos ^2 A\left(1+\sin ^2 A\right)}$
Option 2: $\frac{1}{\sin ^2 A\left(1-\sin ^2 A\right)}$
Option 3: $\frac{1}{\sin ^2 A+\operatorname{cosec}^2 A}$
Option 4: $\frac{1}{\sin ^2 A\left(1+\cos ^2 A\right)}$
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Correct Answer: $\frac{1}{\sin ^2 A\left(1-\sin ^2 A\right)}$
Solution :
$(1+\tan ^2 A)(1+\cot ^2 A)$
= $\sec^2 A × \operatorname{cosec}^2 A$
= $\frac{1}{\cos^2 A} × \frac{1}{\sin^2 A}$
= $\frac{1}{\sin^2 A(1-\sin^2 A)}$
Hence, the correct answer is $\frac{1}{\sin^2 A(1-\sin^2 A)}$.
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