Aptitude Question ID : 94434

If $latex \frac{\tan +\cot }{\tan -\cot } = 2, (0\leq \theta\leq 90^{\circ}),&s=1$ then the value of $latex \sin \theta$ is :
[A]$latex 1$
[B]$latex \frac{1}{2}&s=1$
[C]$latex \frac{\sqrt{3}}{2}&s=1$
[D]$latex \frac{2}{\sqrt{3}}&s=1$

$latex \mathbf{\frac{\sqrt{3}}{2}}&s=1$
$latex \frac{\tan \theta + \cot \theta}{\tan \theta – \cot \theta} = \frac{2}{1}&s=1$
By componendo and dividendo,
$latex \frac{2\tan \theta}{2\cot \theta} = \frac{3}{1}&s=1$
$latex => \frac{\sin \theta }{\cos \theta }\cdot \frac{\sin \theta }{\cos \theta } = 3&s=1$
$latex => \sin ^{2}\theta = 3\cos ^{2}\theta$
$latex => \sin ^{2}\theta = 3 \left ( 1-\sin ^{2}\theta \right )$
$latex => 4\sin ^{2}\theta = 3$
$latex => \sin ^{2}\theta = \frac{3}{4}&s=1$
$latex => \sin \theta = \frac{\sqrt{3}}{2}&s=1$
Hence option [C] is correct answer.


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