Question

If $\mathrm{cosec}\mathrm{\theta }-\cot \mathrm{\theta }=\mathrm{x}$, show that: ​$\mathrm{cosec}\mathrm{\theta }+\cot \mathrm{\theta }=\frac{1}{\mathrm{x}}$ and hence find the values of cos θ and sin ​θ.

Answer 1

Dear student
$$\begin{gathered} Weknowthat, \\ Cose{c}^2\theta -Co{t}^2\theta =1 \\ \Rightarrow \left(Cosec\theta +Cot\theta \right)\left(Cosec\theta -Cot\theta \right)=1...\left(1\right) \\ Andgiventhat\left(Cosec\theta -Cot\theta \right)=x...\left(2\right) \\ Sosubstitutinginequation\left(1\right) \\ \Rightarrow \left(Cosec\theta +Cot\theta \right)x=1 \\ \Rightarrow Cosec\theta +Cot\theta =\frac{1}{x}...\left(3\right) \\ Addingequations\left(2\right)and\left(3\right) \\ \Rightarrow 2Cosec\theta =x+\frac{1}{x} \\ \Rightarrow Cosec\theta =\frac{x^2+1}{2x} \\ \Rightarrow Sin\theta =\frac{2x}{x^2+1}=\frac{P}{H} \\ SObyPythagorastheorem \\ B=\sqrt{H^2-P^2}=\sqrt{{\left(x^2+1\right)}^2-{\left(2x\right)}^2}=\sqrt{x^4+1+2x^2-4x^2}=\sqrt{x^4+1-2x^2}=\sqrt{{\left(x^2-1\right)}^2}=\left(x^2-1\right) \\ So,\cos \theta =\frac{B}{H}=\frac{x^2-1}{x^2+1} \end{gathered}$$
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