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 $1+{\cot }^2\theta =cose{c}^2\theta$
$\therefore \sqrt{1+{\cot }^2\theta }-\cot \theta =q$
$\Rightarrow \sqrt{1+{\cot }^2\theta }=q+\cot \theta$
Squaring both sides, we get
$\therefore \left(1+{\cot }^2\theta \right)={\left(q+\cot \theta \right)}^2$
$\Rightarrow 1+{\cot }^2\theta =q^2+{\cot }^2\theta +2q\cot \theta$
$\Rightarrow 1-q^2=2q\cot \theta$
$\Rightarrow \cot \theta =\frac{1-q^2}{2q}$