Question

Find \( \theta \) from the following equation :

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

Solution

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

\[ \therefore \quad \theta=n\pi \quad \text{or} \quad \theta=\frac{\pi}{4}+n\pi \]