Question

If 3x = cosec θ and $\frac{3}{x}$ = cot θ, than $3\left(x^2-\frac{1}{x^2}\right)=?$
(a) $\frac{1}{27}$
(b) $\frac{1}{81}$
(c) $\frac{1}{3}$
(d) $\frac{1}{9}$

Answer 1

(c) $\frac{1}{3}$
Given: 3x = cosec θ and $\frac{3}{x}$ = cot θ
Also, we can deduce that x = $\frac{\cos ec\theta }{3}$ and $\frac{1}{x}=\frac{cot\theta }{3}$.
So, substituting the values of x and $\frac{1}{x}$ in the given expression, we get:
3$\left(x^2-\frac{1}{x^2}\right)$ = 3$\left({\left(\frac{\cos ec\theta }{3}\right)}^2-{\left(\frac{cot\theta }{3}\right)}^2\right)$
= 3$\left(\left(\frac{\cos e{c}^2\theta }{9}\right)-\left(\frac{co{t}^2\theta }{9}\right)\right)$
= $\frac{3}{9}\left(\cos e{c}^2\theta -co{t}^2\theta \right)$
= $\frac{1}{3}$ [By using the identity: $\left(\cos e{c}^2\theta -co{t}^2\theta =1\right)$]