Question

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

(a) $\frac{1}{27}$ (b) $\frac{1}{81}$ (c) $\frac{1}{3}$ (d) $\frac{1}{9}$

Answer 1

$3x=cosec\theta x=\frac{cosec\theta }{3}{x}^2=\frac{cose{c}^2\theta }{9}----\left(i\right)\frac{3}{x}=cot\theta \frac{1}{x}=\frac{cot\theta }{3}\frac{1}{x^2}=\frac{co{t}^2\theta }{9}----\left(ii\right)3(x^2-\frac{1}{x^2})=3(\frac{cose{c}^2\theta }{9}-\frac{co{t}^2\theta }{9})3\left(\frac{cose{c}^2\theta -co{t}^2\theta }{9}\right)3\left(\frac{1}{9}\right)3\times \frac{1}{9}=\frac{1}{3}$

So answer is $3(x^2-\frac{1}{x^2})=\frac{1}{3}$

(c) $\frac{1}{3}$