Question

If cosec θ – cot θ = 2, then find the value of cosec²θ + cot²θ is ______.

Answer 1

We know that,

$$\begin{gathered} {\mathrm{cosec}}^2\theta -{\cot }^2\theta =1 \\ \Rightarrow \left(\mathrm{cosec}\theta -\cot \theta \right)\left(\mathrm{cosec}\theta +\cot \theta \right)=1 \\ \Rightarrow 2\left(\mathrm{cosec}\theta +\cot \theta \right)=1\left(\mathrm{Given}\right) \\ \Rightarrow \mathrm{cosec}\theta +\cot \theta =\frac{1}{2} \end{gathered}$$
Now,
$$\begin{gathered} {\left(\mathrm{cosec}\theta -\cot \theta \right)}^2+{\left(\mathrm{cosec}\theta +\cot \theta \right)}^2=2^2+{\left(\frac{1}{2}\right)}^2 \\ \Rightarrow {\mathrm{cosec}}^2\theta +{\cot }^2\theta -2\mathrm{cosec}\theta \cot \theta +{\mathrm{cosec}}^2\theta +{\cot }^2\theta +2\mathrm{cosec}\theta \cot \theta =4+\frac{1}{4} \\ \Rightarrow 2\left({\mathrm{cosec}}^2\theta +{\cot }^2\theta \right)=\frac{17}{4} \\ \Rightarrow {\mathrm{cosec}}^2\theta +{\cot }^2\theta =\frac{17}{8} \end{gathered}$$

If cosec θ – cot θ = 2, then find the value of cosec²θ + cot²θ is $\underline{\frac{17}{8}}$.