Question:
If $\operatorname{cosec} \theta-\cot \theta=\alpha$, write the value of $\operatorname{cosec} \theta+\cot \alpha$
Solution:
Given: $\operatorname{cosec} \theta-\cot \theta=\alpha$
We know that, $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$
Therefore,
$\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$
$\Rightarrow(\operatorname{cosec} \theta+\cot \theta)(\operatorname{cosec} \theta-\cot \theta)=1$
$\Rightarrow(\operatorname{cosec} \theta+\cot \theta) \alpha=1$
$\Rightarrow(\operatorname{cosec} \theta+\cot \theta)=\frac{1}{\alpha}$
Hence, $\operatorname{cosec} \theta+\cot \theta=\frac{1}{\alpha}$