Question

If $\text{cosec}\theta -\cot \theta =2,$ then find the value of ${\text{cosec}}^2\theta +{\cot }^2\theta .$

• A. $\frac{8}{15}$
• B. $\frac{15}{8}$
• C. $\frac{8}{17}$
• D. $\frac{17}{8}$

Answer 1

The correct option is D $\frac{17}{8}$

$cosec\theta -\cot \theta =2⟶\left(1\right)$

Now, $(cose{c}^2\theta -{\cot }^2\theta )=1$

$\Rightarrow (cosec\theta -\cot \theta )(cosec\theta +\cot \theta )=1$

$\Rightarrow 2(cosec\theta +\cot \theta )=1$

$\Rightarrow (cosec\theta +\cot \theta )=\frac{1}{2}⟶\left(2\right)$

Adding $\left(1\right)\&\left(2\right),$ we get

$\Rightarrow 2cosec\theta =2+\frac{1}{2}=\frac{5}{2}$

$\Rightarrow cosec\theta =\frac{5}{4}$

$\Rightarrow cose{c}^2\theta =\frac{25}{16}$

Again $\left(2\right)-\left(1\right),$ we get

$\Rightarrow 2\cot \theta =\frac{1}{2}-2=-\frac{3}{2}$

$\Rightarrow \cot \theta =-\frac{3}{4}$

$\Rightarrow {\cot }^2\theta =\frac{9}{16}$

$\Rightarrow cose{c}^2\theta +{\cot }^2\theta =\frac{34}{16}=\frac{17}{8}$