Question

If (cosec θ + cot θ) = m and (cosec θ − cot θ) = n, show that mn = 1.

Answer 1

$$\begin{gathered} \begin{array}{l}\\ \text{We have}(\mathrm{cosec}\theta +\cot \theta )=m...\left(\text{i}\right)\\ \text{Again,}(\mathrm{cosec}\theta -\cot \theta )=n...\left(\text{ii}\right)\\ \text{Now, multiplying}\left(\text{i}\right)\text{and}(\text{ii), we get:}\end{array} \\ \begin{array}{l}(\mathrm{cosec}\theta +\cot \theta )\times (\mathrm{cosec}\theta -\cot \theta )=mn\\ =>\cos {\text{ec}}^2\theta -{\cot }^2\theta =mn\\ =>1=mn\text{[}\because \cos {\text{ec}}^2\theta -{\cot }^2\theta =1]\\ \therefore mn=1\end{array} \end{gathered}$$