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Q)

Find the value of $\tan\large\frac{\pi}{8}$

$(a)\;\sqrt 2\qquad(b)\;0\qquad(c)\;\large\frac{1}{\sqrt 2}$$\qquad(d)\;\sqrt 2-1$

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A)
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Let $\theta=\large\frac{\pi}{8}$
$2\theta=\large\frac{\pi}{4}$
$\tan 2\theta=\large\frac{2\tan\theta}{1-\tan^2\theta}$
$\tan\large\frac{\pi}{4}=\frac{2\tan\Large\frac{\pi}{8}}{1-\tan^2\Large\frac{\pi}{8}}$
Let $x=\tan\large\frac{\pi}{8}$
$I=\large\frac{2x}{1-x^2}$
$1-x^2=2x$
$x^2+2x-1=0$
$x=\large\frac{-2\pm 2\sqrt 2}{2}$
$\Rightarrow -1+\sqrt 2,-1-\sqrt 2$
$\tan\large\frac{\pi}{8}$$=\sqrt 2-1$
Hence (d) is the correct answer.
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