If $3x=cosec\theta$ and $\frac{3}{x}\ =\cot\theta$, find the value of $3\left( x^{2} -\frac{1}{x^{2}}\right)$.

Given: $3x=cosec\theta$ and $\frac{3}{x}=cot\theta$.

To do: To find out the value of $3\left( x^{2} -\frac{1}{x^{2}}\right)$.

Solution:
Here given $3x=cosec\theta$ and $\frac{3}{x}=cot\theta$

Then

$( 3x)^{2} -\left(\frac{3}{x}\right)^{2} =cosec^{2} θ-cot^{2} θ$

$\Rightarrow 9x^{2} -\frac{9}{x^{2}} \ =1$ $\left( as\ known\ cosec^{2} \theta -cot^{2} \theta =1\right)$   

$\Rightarrow 9\left( x^{2} -\frac{1}{x^{2}}\right) =1\$

$\Rightarrow 3\left( x^{2} -\frac{1}{x^{2}}\right) =\frac{1}{3}\$

Hence the value of $3\left( x^{2} -\frac{1}{x^{2}}\right)$ is $\ \frac{1}{3} $.
 

Updated on: 10-Oct-2022

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