If cosec2 θ (1 + cos θ) (1 − cos θ) = λ, then find the value of λ.

Given:

$\operatorname{cosec}^{2} \theta(1+\cos \theta)(1-\cos \theta)=\lambda$

$\Rightarrow \operatorname{cosec}^{2} \theta\{(1+\cos \theta)(1-\cos \theta)\}=\lambda$

$\Rightarrow \quad \operatorname{cosec}^{2} \theta\left(1-\cos ^{2} \theta\right)=\lambda$

$\Rightarrow \quad \operatorname{cosec}^{2} \theta \sin ^{2} \theta=\lambda$

$\Rightarrow \quad \frac{1}{\sin ^{2} \theta} \times \sin ^{2} \theta=\lambda$

$\Rightarrow \quad 1=\lambda$

$\Rightarrow \quad \lambda=1$