Question

If $cosec \theta - \cot \, \theta = q$, then the value of $\cot \, \theta$ is:

• A. $\frac{1 - q^2}{2q}$
• B. $\frac{1 + q^2}{q}$
• C. $\frac{q}{1 - q^2}$
• D. $\frac{q}{1 + q^2}$

Answer 1

Answer: (A)

Given : $cosec \, \theta - \cot \, \theta = q$
We know that $1 + \cot^2 \theta = cosec^2 \theta$
$\therefore \, \sqrt{1 + \cot^2 \theta} - \cot \theta = q $
$\Rightarrow \, \, \sqrt{1 + \cot^2 \theta} = q + \cot \theta $
Squaring both sides, we get
$\therefore \, \left(1+ \cot^{2} \theta\right) = \left(q + \cot \theta\right)^{2}$
$ \Rightarrow 1 + \cot^{2} \theta = q^{2} + \cot^{2} \theta+ 2 q \cot\theta $
$\Rightarrow 1 - q^{2} = 2q \cot\theta $
$\Rightarrow \cot \theta = \frac{ 1- q^{2}}{2q}$