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

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

Trigonometric Functions

Solution:

$\alpha+\beta=\frac{\pi}{2}$ or $\alpha=\frac{\pi}{2}-\beta$
$\Rightarrow \tan \alpha=\cot \beta$
or $\tan \alpha \tan \beta=1$ ...(i)
Again, $\beta+\gamma=\alpha$ or $\gamma=\alpha-\beta$
$\Rightarrow \tan \gamma=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}$
$=\frac{\tan \alpha-\tan \beta}{2}$ [using Eq. (i)]
or $\tan \alpha=\tan \beta+2 \tan \gamma$