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If sin2 θ cos2 θ (1 + tan2 θ) (1 + cot2 θ) = λ, then find the value of λ.

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 .

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