Question

If $4\sin \theta =3$, find the value of $x$, if $\sqrt{\frac{{\text{cosec}}^2\theta -{\cot }^2\theta }{{\sec }^2\theta -1}}+2\cot \theta =\frac{\sqrt{7}}{x}+\cos \theta$.

Answer 1

$\sin \theta =\frac{3}{4}\Rightarrow \csc \theta =\frac{4}{3}$

$\Rightarrow \cos \theta =\frac{5}{4}\Rightarrow \sec \theta =\frac{4}{5}$

$\Rightarrow \cot \theta =\frac{5}{3}$

$\therefore \sqrt{\frac{{\csc }^2\theta -{\cot }^2\theta }{{\sec }^2\theta -1}}+2\cot \theta =\frac{\sqrt{7}}{x}+\cos \theta$

$\therefore \sqrt{\frac{{\left(\frac{4}{3}\right)}^2-{\left(\frac{5}{3}\right)}^2}{{\left(\frac{4}{5}\right)}^2-1}}+2\left(\frac{5}{3}\right)=\frac{\sqrt{7}}{x}+\left(\frac{5}{4}\right)$

$\sqrt{\frac{25}{9}}+2\left(\frac{5}{3}\right)-\left(\frac{5}{4}\right)=\frac{\sqrt{7}}{x}$

$\frac{5}{3}+\frac{10}{3}-\frac{5}{4}=\frac{\sqrt{7}}{x}$

$=\frac{45}{12}=\frac{\sqrt{7}}{x}$

$x=\frac{12\sqrt{7}}{45}$