Question

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

Answer 1

(d) cos 2β

$$\begin{gathered} \text{We have:} \\ \cos (\alpha +\beta )=0 \\ =>\cos (\alpha +\beta )=\cos {90}^0 \\ =>\alpha +\beta ={90}^0 \\ =>\alpha ={90}^0-\beta ...\left(\text{i}\right) \\ \text{Now, sin}(\alpha -\beta ) \\ =\sin [({90}^0-\beta )-\beta ]\left[\mathrm{U}\text{sing}\left(\text{i}\right)\right] \\ =\sin ({90}^0-2\beta ) \\ =\cos 2\beta [\because \sin ({90}^0-\theta )=\cos \theta ] \end{gathered}$$