Question

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

Answer 1

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

$\Rightarrow \sqrt{\frac{1}{{\tan }^2\theta }}+2\cot \theta =\frac{\sqrt{7}}{x}+\cos \theta$

$\Rightarrow \cot \theta +2\cot \theta =\frac{\sqrt{7}}{x}+\cos \theta$

Since, $\sin \theta =\frac{3}{4};\cot \theta =\frac{\sqrt{7}}{x}$ and $\cos \theta =\frac{\sqrt{7}}{4}$

$\Rightarrow 3\left(\frac{\sqrt{7}}{3}\right)=\frac{\sqrt{7}}{x}+\frac{\sqrt{7}}{4}$

$\Rightarrow 3\left(\frac{\sqrt{7}}{4}\right)=\frac{\sqrt{7}}{x}$

$\Rightarrow x=4/3$