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  4. If α+β=(π/2) and β+γ, then tan α equals

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

IIT JEEIIT JEE 2001

Solution:

Given, $ \alpha+\beta=\pi/2 \Rightarrow \alpha=(\pi/2)=\beta$
$\Rightarrow tan \alpha=tan(\pi/2-\beta)$
$\Rightarrow tan \alpha=cot \beta \Rightarrow tan \alpha tan \beta=1$
Again, $\beta+gamma=\alpha [given]$
$\Rightarrow \gamma=(\alpha-\beta)$
$\Rightarrow tan \gamma=tan (\alpha-\beta$
$\Rightarrow tan \gamma=\frac{tan \alpha-tan\beta}{1+tan \alpha tan\beta}$
$\Rightarrow \frac{tan \alpha-tan\beta}{1+1}$
$\therefore 2 tan \gamma = tan \alpha - tan \beta$
$\Rightarrow tan \alpha=tan \beta+2 tan \gamma$