Question

If $3x$ = $cosec\theta$ and $\frac{3}{x}$ = $cot\theta$ then find $3(x^2+\frac{1}{x^2})$

Answer 1

$3x=\csc \theta$ and $\frac{3}{x}=\cot \theta$

$\Rightarrow x=\frac{\csc \theta }{3}$ and $\frac{1}{x}=\frac{\cot \theta }{3}$

$3(x^2+\frac{1}{x^2})=3({\left(\frac{\csc \theta }{3}\right)}^2+\frac{1}{{\left(\frac{\cot \theta }{3}\right)}^2})$

$=3(\frac{{\csc }^2\theta }{9}+\frac{9}{{\cot }^2\theta })$

$=3(\frac{1+{\cot }^2\theta }{9}+\frac{9}{{\cot }^2\theta })$

$=3\left(\frac{{\cot }^4\theta +{\cot }^2\theta +81}{9{\cot }^2\theta }\right)$

$=\left(\frac{{\cot }^4\theta +{\cot }^2\theta +81}{3{\cot }^2\theta }\right)$

$=\frac{{\cot }^2\theta ({\cot }^2\theta +1)+81}{3{\cot }^2\theta }$

$=\frac{{\cot }^2\theta {\csc }^2\theta +81}{3{\cot }^2\theta }$

$=\frac{{\cot }^2\theta {\csc }^2\theta +81}{3{\cot }^2\theta }$