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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions1 Mark
Question
If $\cot(\alpha+\beta)=0,$ then write the value of $\sin(\alpha+2\beta).$
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Answer

$\cot(\alpha+\beta)=0$
$\frac{\cos(\alpha+\beta)}{\sin(\alpha+\beta)}=0$
$\therefore\cos(\alpha+\beta)=0$
$\therefore\cos\alpha\cos\beta=\sin\alpha\sin\beta$
$\sin(\alpha+2\beta)$
$=\sin\alpha\cos2\beta+\cos\alpha\sin2\beta$
$=\sin\alpha(\cos^2\beta-\sin^2\beta)+\cos\alpha(2\sin\beta\cos\beta)$
$=\sin\alpha(\cos^2\beta-\sin^2\beta)+2\sin^2\beta(\sin\alpha\sin\beta)$
$=\sin\alpha(\cos^2\beta-\sin^2\beta+2\sin^2\beta)$
$=\sin\alpha(\cos^2\beta+\sin\beta)$
$=\sin\alpha$
$\cos(\alpha+\beta)=0$
$\Rightarrow\alpha+\beta=\frac\pi2$
$\Rightarrow\alpha=\frac\pi2-\beta$
$\therefore\sin(\alpha+2\beta)=\sin\alpha$
$\sin(\alpha+2\beta)=\sin\Big(\frac\pi2-\beta\Big)=\cos\beta$
$\therefore\sin(\alpha+2\beta)=\sin\alpha \text{ or }\cos\beta$

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