Question

The sum of three positive numbers $\alpha, \beta, \gamma$ is equal to $\frac{\pi}{2}$. If $e ^{\cot \alpha}, e ^{\cot \beta}$ and $e ^{\cot \gamma}$ form a G.P., then which of the following hold(s) good?

• A. $\sum \cot \alpha=\prod \cot \alpha $
• B. $2 \cot \beta=\cot \alpha+\cot \gamma$
• C. $ \cot \alpha \cot \gamma=3 $
• D. $4 \prod \cos \alpha=\sum \sin 2 \alpha$

Answer 1

Answer: (A), (B), (C), (D)

$\alpha+\beta=\frac{\pi}{2}-\gamma$
$\cot (\alpha+\beta)=\cot \left(\frac{\pi}{2}-\gamma\right)=\tan \gamma $
$\frac{\cot \alpha \cot \beta-1}{\cot \beta+\cot \alpha}=\frac{1}{\cot \gamma}$
$\cot \alpha \cot \beta \cot \gamma=\cot \alpha+\cot \beta+\cot \gamma$(A)
Now $e ^{\cot \alpha}, e ^{\cot \beta}, e ^{\cot \gamma}$ in G.P.
$\therefore \cot \alpha, \cot \beta, \cot \gamma$ in A.P.
$\cot \alpha+\cot \gamma=2 \cot \beta \Rightarrow $ (B)
From (1)
$\cot \alpha \cot \beta \cot \gamma=3 \cot \beta \Rightarrow$(C)
$\cot \beta \neq 0 \Rightarrow \cot \alpha \cdot \cot \gamma=3$
(As $\alpha, \beta, \gamma>0$ )
$\text { Also } \prod \cos \alpha =\frac{1}{2} \cos \gamma(2 \cos \alpha \cos \beta)=\frac{1}{2} \cos \gamma[\cos (\alpha+\beta)+\cos (\alpha-\beta)] $
$ =\frac{1}{2} \cos \gamma[\sin \gamma+\cos (\alpha-\beta)]=\frac{1}{4} \sin 2 \gamma+\frac{1}{4}[2 \cos \gamma \cos (\alpha-\beta)]$
$ =\frac{1}{4} \sin 2 \gamma+\frac{1}{4}[\cos (\gamma+\alpha-\beta)+\cos (\gamma+\beta-\alpha)] $
$ =\frac{1}{4} \sin 2 \gamma+\frac{1}{4}\left[\cos \left(\frac{\pi}{2}-2 \beta\right)+\cos \left(\frac{\pi}{2}-2 \alpha\right)\right] $
$ =\frac{1}{4} \sin 2 \gamma+\frac{1}{4}[\sin 2 \beta+\sin 2 \alpha] $
$\Rightarrow \prod \cos \alpha =\frac{1}{4}\left[\sum \sin 2 \alpha\right]$
Hence $4 \prod \cos \alpha=\sum \sin 2 \alpha \Rightarrow$ (D)