If $\mathrm{cosec}\left(\mathrm{\theta }-\mathrm{\alpha }\right),\mathrm{cosec\theta },\mathrm{cosec}\left(\mathrm{\theta }+\mathrm{\alpha }\right)$ are in A.P. where $\mathrm{\alpha }\in \left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ then the number of values of α  is

$\frac{1}{\sin \left(\mathrm{\theta }-\mathrm{\alpha }\right)}+\frac{1}{\sin \left(\mathrm{\theta }+\mathrm{\alpha }\right)}=\frac{2}{\mathrm{sin\theta }}$
$\begin{array}{l}\Rightarrow {\sin }^2\mathrm{\theta cos}\mathrm{\alpha }={\sin }^2\mathrm{\theta }-{\sin }^2\mathrm{\alpha }\\ (1-\cos \mathrm{\alpha }){\sin }^2\mathrm{\theta }={\sin }^2\mathrm{\alpha }=1-{\cos }^2\mathrm{\alpha }\\ \Rightarrow (1-\cos \mathrm{\alpha })\left\{{\sin }^2\mathrm{\theta }-(1+\cos \mathrm{\alpha })\right\}=0\end{array}$
$\overline{)\begin{array}{c}\Rightarrow \mathrm{cos\alpha }=1\\ \Rightarrow \mathrm{\alpha }=0\\ \\ \end{array}}\begin{array}{l}\xcancel{{\sin }^2\mathrm{\theta }=1+\mathrm{cos\alpha }}\\ \because {\sin }^2\mathrm{\theta }>1 \mathrm{as}\\ \mathrm{cos\alpha }>0 \forall \mathrm{\alpha }\in \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)\end{array}$