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)
$\cot \beta \ne 0\Rightarrow \cot \alpha \cdot \cot \gamma =3$
(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)