Question

If $3x=\csc \theta$ and $\frac{3}{x}=\cot \theta$ find the value of $3(x^2-\frac{1}{x^2})$

Answer 1

If $3x=cosec\theta$ and $\frac{3}{x}=-\cot \theta$

Find the value of $3(x^2-\frac{1}{x^2})$

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

$=\frac{1}{3}(\frac{1}{{\sin }^2\theta }-\frac{{\cos }^2\theta }{{\sin }^2\theta })$

$=\frac{1}{3}\left(\frac{1-{\cos }^2\theta }{{\sin }^2\theta }\right)$

$=\frac{1}{3}\left(\frac{{\sin }^2\theta }{{\cos }^2\theta }\right)$

$=\frac{1}{3}$