Show that $\tan {\pi \over 8} = \sqrt 2 - 1$, using the identity $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\theta }}$

Using $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\theta }}$ with $\theta = {\pi \over {16}}$:

\eqalign{ & \tan {\pi \over 8} = {{2\tan {\pi \over {16}}} \over {1 - {{\tan }^2}{\pi \over {16}}}} \cr & \tan {\pi \over 8} = 0.41421.... \cr}

I essentially get an answer equivalent to $\sqrt 2 - 1$ but not an "exact" answer, how do I go about this question to get a result in "exact" form?

Thank you.

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algebra-precalculus trigonometry
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asked Apr 4 "13 at 23:26

seekerseeker
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We have $\tan(2 \theta) = \dfrac{2 \tan(\theta)}{1-\tan^2(\theta)}$. Take $\theta = \dfrac{\pi}8$ and let $t = \tan(\pi/8)$. We then get$$\tan(\pi/4) = \dfrac{2 \tan(\pi/8)}{1-\tan^2(\pi/8)} = \dfrac{2t}{1-t^2}$$Hence, we get$$1-t^2 = 2t \implies (t+1)^2 = 2 \implies t = -1 \pm \sqrt2$$Since $t > 0$, we get that $\tan(\pi/8) = \sqrt2-1$

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answered Apr 4 "13 at 23:30
user17762user17762
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You"re going about this the wrong way; instead, set $\theta = \frac{\pi}{8}$. You get:

$$\tan \frac{\pi}{4} = 1 = \frac{2\tan \frac{\pi}{8}}{1-\tan^2 \frac{\pi}{8}}$$

Multiply through by $1-\tan^2 \frac{\pi}{8}$ and solve the resulting quadratic equation.

If it helps simplify notation, substitute $t=\tan^2 \frac{\pi}{8}$. The equation has two roots, one positive and one negative, so you"ll need to give reasoning about which to pick.

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answered Apr 4 "13 at 23:30

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An identity that is useful when considering the stereographic projection and the half-tangent integration substitution:$$\tan(x/2)=\frac{\sin(x)}{1+\cos(x)}$$We know that $\sin(\pi/4)=\cos(\pi/4)=1/\sqrt2$. Therefore,\begin{align}\tan(\pi/8)&=\frac{1/\sqrt2}{1+1/\sqrt2}\frac{1-1/\sqrt2}{1-1/\sqrt2}\\&=\sqrt{2}-1\end{align}

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answered Apr 4 "13 at 23:48

robjohn♦robjohn
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$$\sqrt2-1=\csc\dfrac\pi4-\cot\dfrac\pi4=\dfrac{1-\cos\dfrac\pi4}{\sin\dfrac\pi4}=\dfrac{2\sin^2\dfrac\pi8}{2\sin\dfrac\pi8\cos\dfrac\pi8}=?$$ using $\sin2A=2\sin A\cos A,\cos2B=1-2\sin^2B$

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answered Oct 23 "19 at 7:12

lab bhattacharjeelab bhattacharjee
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