Question

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

Answer 1

We have,

$\cot \theta =\frac{3}{x}$

$\cos ec\theta =3x$

Since, $3(x^2-\frac{1}{x^2})=?$

We know that

$\cos e{c}^2\theta -{\cot }^2\theta =1$

Therefore,

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

$9x^2-\frac{9}{x^2}=1$

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

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

Hence, the value is $\frac{1}{3}$.