Question
If $\text{cosec }\theta-\cot\theta=\alpha,$ write the value of $\text{cosec }\theta+\cos\alpha$.
✓
Answer
Given, $\text{cosec }\theta-\cot\theta=\alpha$
We know that, $\text{cosec}^2\theta-\cot^2\theta=1$
Therefore,
$\text{cosec}^2\theta-\cot^2\theta=1$
$\Rightarrow\ (\text{cosec }\theta+\cot\theta)(\text{cosec }\theta-\cot\theta)=1$
$\Rightarrow\ (\text{cosec }\theta+\cot\theta)\alpha=1$
$\Rightarrow\ (\text{cosec }\theta+\cot\theta)=\frac{1}{\alpha}$
Hence, $\text{cosec }\theta-\cot\theta=\frac{1}{\alpha}$
We know that, $\text{cosec}^2\theta-\cot^2\theta=1$
Therefore,
$\text{cosec}^2\theta-\cot^2\theta=1$
$\Rightarrow\ (\text{cosec }\theta+\cot\theta)(\text{cosec }\theta-\cot\theta)=1$
$\Rightarrow\ (\text{cosec }\theta+\cot\theta)\alpha=1$
$\Rightarrow\ (\text{cosec }\theta+\cot\theta)=\frac{1}{\alpha}$
Hence, $\text{cosec }\theta-\cot\theta=\frac{1}{\alpha}$
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