If cot (α + β) = 0,

Question:

If cot (α + β) = 0, then write the value of sin (α + 2β).

Solution:

$\cot (\alpha+\beta)=0$

$\Rightarrow \alpha+\beta=\frac{\pi}{2}$       (1)

$\beta=\frac{\pi}{2}-\alpha$                            (2)  

$\alpha=\frac{\pi}{2}-\beta$                             (3)

Now, $\sin (\alpha+2 \beta)=\sin (\alpha+\beta+\beta)$

$=\sin \left(\frac{\pi}{2}+\frac{\pi}{2}-\alpha\right)$

$=\sin (\pi-\alpha)$

$=\sin \alpha$

Now, $\sin (\alpha+2 \beta)=\sin (\alpha+2 \beta)$

$=\sin \left(\frac{\pi}{2}-\beta+2 \beta\right)$

$=\sin \left(\frac{\pi}{2}+\beta\right)$

$=\cos \beta$

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