Question

If $\cos (\alpha +\beta )=0$, then $\sin (\alpha +2\beta )=$

• A. $\sin \alpha$
• B. $-\sin \alpha$
• C. $\cos \beta$
• D. $-\cos \beta$

Answer 1

Answer: (A), (C)

The correct options are
A $\sin \alpha$
C $\cos \beta$

$\cos (\alpha +\beta )=0\Rightarrow \alpha +\beta ={90}^o$ $\Rightarrow \alpha ={90}^o-\beta \Rightarrow \beta ={90}^o-\alpha$

Now, $\sin (\alpha +2\beta )=\sin (\alpha +2({90}^o-\alpha \left)\right)=\sin ({180}^o-\alpha )=\sin \alpha$

Similarly, $\sin (\alpha +2\beta )=\sin ({90}^o-\beta +2\beta )=\sin ({90}^o+\beta )=\cos \beta$

Therefore, Answer are $\sin \alpha ,\cos \beta$