Question

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

Answer 1

$3x=\csc \theta ,\frac{3}{y}=\cot \theta$

$\Rightarrow 9x^2={\csc }^2\theta ,\frac{9}{y^2}={\cot }^2\theta$

$\Rightarrow 9x^2-\frac{9}{y^2}={\csc }^2\theta -{\cot }^2\theta$

$\Rightarrow 9x^2-\frac{9}{y^2}=1$ since ${\csc }^2\theta -{\cot }^2\theta =1$

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

Dividing both sides by $3$

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