Question:
If $\sin ^{2} \theta \cos ^{2} \theta\left(1+\tan ^{2} \theta\right)\left(1+\cot ^{2} \theta\right)=\lambda$, then find the value of $\lambda .$
Solution:
Given:
$\sin ^{2} \theta \cos ^{2} \theta\left(1+\tan ^{2} \theta\right)\left(1+\cot ^{2} \theta\right)=\lambda$
$\Rightarrow \quad \sin ^{2} \theta \cos ^{2} \theta \sec ^{2} \theta \operatorname{cosec}^{2} \theta=\lambda$
$\Rightarrow \quad\left(\sin ^{2} \theta \operatorname{cosec}^{2} \theta\right) \times\left(\cos ^{2} \theta \sec ^{2} \theta\right)=\lambda$
$\Rightarrow\left(\sin ^{2} \theta \times \frac{1}{\sin ^{2} \theta}\right)\left(\cos ^{2} \theta \times \frac{1}{\cos ^{2} \theta}\right)=\lambda$
$\Rightarrow \lambda=1 \times 1=1$
Hence, the value of $\lambda$ is 1 .