Mathematics - Trigonometric Equation Question with Solution | TestHub

Given, $\begin{array}{rl}& \cot x=-\frac{5}{\sqrt{11}}\\ & \frac{1-{\tan }^2\frac{x}{2}}{2\tan \frac{x}{2}}=-\frac{5}{\sqrt{11}}\\ & \tan \frac{x}{2}=\sqrt{11},-\frac{1}{\sqrt{11}}\\ & \text{So, }\tan \frac{x}{2}=\sqrt{11},\text{ As }\frac{\pi }{4}<\frac{x}{2}<\frac{\pi }{2}\end{array}$ $\begin{array}{rl}& \text{As, }x\in \left(\frac{\pi }{2},\pi \right)\\ & \text{Now, }\left(\sin \frac{11x}{2}\right)(\sin 6x-\cos 6x)+\left(\cos \frac{11x}{2}\right)(\sin 6x+\cos 6x)\\ & =\left\{\sin 6x\sin \frac{11x}{2}+\cos \frac{11x}{2}\cos 6x\right\}\\ & =\cos \left(6x-\frac{11x}{2}\right)+\sin \left(6x-\frac{11x}{2}\right)\\ & =\cos \frac{x}{2}+\sin \frac{x}{2}\\ & =\frac{1}{2\sqrt{3}}+\frac{\sqrt{11}}{2\sqrt{3}}\\ & =\frac{\sqrt{11}+1}{2\sqrt{3}}\Rightarrow \text{ Option (2) is correct. }\end{array}$