If 3x = cosec θ and


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: $\left(\cos e c^{2} \theta-\cot ^{2} \theta=1\right)$ ]


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