If cos , = 2 , , , , - , ,1 2 , , - , , , then tan 2 cot 2 has th… - Vidyadip

MCQ

If $cos\, \alpha = \frac{{2\,\cos \,\beta \,\, - \,\,1}}{{2\,\, - \,\,\cos \,\beta }}$ then $tan \frac{\alpha}{2}$ $cot \frac{\beta}{2}$ has the value equal to, where $(0 < \alpha < \pi$ and $0 < \beta < \pi$)

A
$2$
B
$\sqrt 2 \,$
C
$3$
✓
$\sqrt 3 \,$

✓

Answer

Correct option: D.

$\sqrt 3 \,$

d
$\frac{1}{{\cos \,\alpha }} = \frac{{2\,\, - \,\,\cos \,\beta }}{{2\,\cos \,\beta \,\, - \,\,1}}$

Applying $C/D$

$ \Rightarrow = \frac{{1\,\, - \,\,\cos \,\alpha }}{{1\,\, + \,\,\cos \,\alpha }} = \frac{{3\,\left( {1\,\, - \,\,\cos \,\beta } \right)}}{{1\,\, + \,\,\cos \,\beta }}$

$\Rightarrow\, tan^2 \frac{\alpha}{2} = 3 tan^2 \frac{\beta}{2} \Rightarrow tan^2 \frac{\alpha}{2} \,cot^2 \frac{\beta}{2}= 3$

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