Question

If cot (α + β) = 0, then write the value of sin (α + 2β).

Answer 1

$$\begin{gathered} \cot \left(\alpha +\beta \right)=0 \\ \Rightarrow \alpha +\beta =\frac{\mathrm{\pi }}{2}\left(1\right) \\ \beta =\frac{\mathrm{\pi }}{2}-\alpha \left(2\right) \\ \mathrm{\alpha }=\frac{\mathrm{\pi }}{2}-\mathrm{\beta }\left(3\right) \\ \mathrm{Now},\sin \left(\mathrm{\alpha }+2\mathrm{\beta }\right)=\sin \left(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\beta }\right) \\ =\sin \left(\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}-\mathrm{\alpha }\right) \\ =\sin \left(\mathrm{\pi }-\mathrm{\alpha }\right) \\ =\sin \mathrm{\alpha } \\ \mathrm{Now},\sin \left(\mathrm{\alpha }+2\mathrm{\beta }\right)=\sin \left(\mathrm{\alpha }+2\mathrm{\beta }\right) \\ =\sin \left(\frac{\mathrm{\pi }}{2}-\mathrm{\beta }+2\mathrm{\beta }\right) \\ =\sin \left(\frac{\mathrm{\pi }}{2}+\mathrm{\beta }\right) \\ =\cos \mathrm{\beta } \end{gathered}$$