If α + β = frac{π }{2} and β + gamma = α , then tan ,α equals

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3.Trigonometrical Ratios, Functions and Identities

medium

If $\alpha + \beta = \frac{\pi }{2}$ and $\beta + \gamma = \alpha ,$ then $\tan \,\alpha $ equals

$2\,(\tan \beta + \tan \gamma )$

$\tan \beta + \tan \gamma $

$\tan \beta + 2\,\tan \gamma $

$2\,\tan \beta + \tan \gamma $

(IIT-2001)

Solution

(c)$\alpha + \beta = \frac{\pi }{2} \Rightarrow \tan \beta = \cot \alpha $
$\tan (\beta + \gamma ) = \tan \alpha $ ==> $\tan \alpha = \frac{{\tan \beta + \tan \gamma }}{{1 – \tan \beta \tan \gamma }}$
==> $\tan \alpha = \frac{{\cot \alpha + \tan \gamma }}{{1 – \cot \alpha \tan \gamma }}$
==> $\tan \alpha – \tan \gamma = \cot \alpha + \tan \gamma $
==> $\tan \alpha = \tan \beta + 2\tan \gamma $.

Standard 11

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