If $3 x=\operatorname{cosec} \theta$ and $\frac{3}{x}=\cot \theta$, than $3\left(x^{2}-\frac{1}{x^{2}}\right)=?$
(a) $\frac{1}{27}$
(b) $\frac{1}{81}$
(c) $\frac{1}{3}$
(d) $\frac{1}{9}$
(c) $\frac{1}{3}$
Given: $3 x=\operatorname{cosec} \theta$ and $\frac{3}{x}=\cot \theta$
Also, we can deduce that $x=\frac{\cos e c \theta}{3}$ and $\frac{1}{x}=\frac{\cot \theta}{3}$.
So, substituting the values of $x$ and $\frac{1}{x}$ in the given expression, we get:
$3\left(x^{2}-\frac{1}{x^{2}}\right)=3\left(\left(\frac{\operatorname{cosec} \theta}{3}\right)^{2}-\left(\frac{\cot \theta}{3}\right)^{2}\right)$
$=3\left(\left(\frac{\operatorname{cosec}^{2} \theta}{9}\right)-\left(\frac{\cot ^{2} \theta}{9}\right)\right)$
$=\frac{3}{9}\left(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta\right)$
$=\frac{1}{3} \quad$ [By using the identity: $\left(\cos e c^{2} \theta-\cot ^{2} \theta=1\right)$ ]