Question

Find \(x\) from the following equation :

\[ x\cot\left(\frac{\pi}{2}+\theta\right) + \tan\left(\frac{\pi}{2}+\theta\right)\sin\theta + \cosec\left(\frac{\pi}{2}+\theta\right) = 0 \]

Solution

\[ \begin{aligned} &x\cot\left(\frac{\pi}{2}+\theta\right) + \tan\left(\frac{\pi}{2}+\theta\right)\sin\theta + \cosec\left(\frac{\pi}{2}+\theta\right) = 0 \\[8pt] \Rightarrow\;& -x\tan\theta-\cot\theta\sin\theta+\sec\theta=0 \\[8pt] \Rightarrow\;& -x\tan\theta-\cos\theta+\sec\theta=0 \\[8pt] \Rightarrow\;& x\tan\theta=\sec\theta-\cos\theta \\[8pt] \Rightarrow\;& x= \frac{\sec\theta-\cos\theta}{\tan\theta} \\[8pt] \Rightarrow\;& x= \frac{\frac1{\cos\theta}-\cos\theta}{\frac{\sin\theta}{\cos\theta}} \\[8pt] \Rightarrow\;& x= \frac{1-\cos^2\theta}{\sin\theta} \\[8pt] \Rightarrow\;& x= \frac{\sin^2\theta}{\sin\theta} \\[8pt] \Rightarrow\;& x=\sin\theta \end{aligned} \]

Answer :

\[ \boxed{x=\sin\theta} \]