Question

If cos (α + β) = 0 then sin (α – β) = ?
(a) sin 2α
(b) cos 2β
(c) sin α
(d) cos β

Answer 1

Given: cos(α + β) = 0

$$\begin{gathered} \mathrm{As}\mathrm{we}\mathrm{know}\mathrm{that}, \\ \cos 90°=0 \\ \mathrm{Since},\cos \left(\alpha +\beta \right)=0 \\ \Rightarrow \alpha +\beta =90° \\ \Rightarrow \alpha =90°-\beta ...\left(1\right) \\ \mathrm{Now}, \\ \sin \left(\alpha -\beta \right)=\sin \left(90°-\beta -\beta \right) \\ =\sin \left(90°-2\beta \right) \\ =\cos 2\beta \left(\because \sin \left(90°-\theta \right)=\cos \theta \right) \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{b}\right). \end{gathered}$$