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