Question

If cosec θ + cot θ = x, find the value of cosec θ – cot θ.

Answer 1

Dear Student,
Here is the solution of your asked query:
$$\begin{gathered} \mathrm{Given},\mathrm{cosec\theta }+\mathrm{cot\theta }=\mathrm{x} \\ \mathrm{We}\mathrm{know}\mathrm{that}{\mathrm{cosec}}^2\mathrm{\theta }-{\cot }^2\mathrm{\theta }=1 \\ \Rightarrow \left(\mathrm{cosec\theta }+\mathrm{cot\theta }\right)\left(\mathrm{cosec\theta }-\mathrm{cot\theta }\right)=1\{\mathrm{using}\mathrm{the}\mathrm{identity}{\mathrm{a}}^2-{\mathrm{b}}^2=\left(\mathrm{a}-\mathrm{b}\right)\left(\mathrm{a}+\mathrm{b}\right)\} \\ \Rightarrow \mathrm{x}\left(\mathrm{cosec\theta }-\mathrm{cot\theta }\right)=1 \\ \Rightarrow \mathrm{cosec\theta }-\mathrm{cot\theta }=\frac{1}{\mathrm{x}} \end{gathered}$$
Regards