If ( + )=0, then write the value of ( +2 ). - Vidyadip

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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