Question
If $\text{cosec}^2\theta(1+\cos\theta)(1-\cos\theta)=\lambda,$ then find is the value of $\lambda$.
✓
Answer
$\text{cosec}^2\theta(1+\cos\theta)(1-\cos\theta)=\lambda$
$\Rightarrow\ \text{cosec}^2\theta(1-\cos^2\theta)=\lambda \ \{(\text{a}+\text{b})(\text{a}-\text{b})=\text{a}^2-\text{b}^2\}$
$\Rightarrow\ \text{cosec}^2\theta\text{ x}\sin\theta=\lambda \ (1-\cos^2\theta=\sin^2\theta)$
$\Rightarrow\ 1=\lambda \ (\sin\theta\text{ cosec }\theta=1)$
$\therefore\ \lambda=1$
$\Rightarrow\ \text{cosec}^2\theta(1-\cos^2\theta)=\lambda \ \{(\text{a}+\text{b})(\text{a}-\text{b})=\text{a}^2-\text{b}^2\}$
$\Rightarrow\ \text{cosec}^2\theta\text{ x}\sin\theta=\lambda \ (1-\cos^2\theta=\sin^2\theta)$
$\Rightarrow\ 1=\lambda \ (\sin\theta\text{ cosec }\theta=1)$
$\therefore\ \lambda=1$
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