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pishuonlain [190]
4 years ago
9

I'm confused! Please explain!

Mathematics
1 answer:
iren [92.7K]4 years ago
4 0

First, use the fact that \tan x is \pi-periodic. This means \tan(x+\pi)=\tan x for all x, and in particular

\tan\dfrac{9\pi}8=\tan\left(\dfrac\pi8+\pi\right)=\tan\dfrac\pi8

Now, recall the half-angle identities,

\sin^2\dfrac x2=\dfrac{1-\cos x}2

\cos^2\dfrac x2=\dfrac{1+\cos x}2

\implies\tan^2\dfrac x2=\dfrac{\sin^2\frac x2}{\cos^2\frac x2}=\dfrac{1-\cos x}{1+\cos x}

When we take the square root, we get two possible values, but we know \tan x>0 for 0, so we know to take the positive square root:

\tan\dfrac x2=\sqrt{\dfrac{1-\cos x}{1+\cos x}}

So we have

\tan\dfrac\pi8=\sqrt{\dfrac{1-\cos\frac\pi4}{1+\cos\frac\pi4}}=\sqrt{\dfrac{1-\frac1{\sqrt2}}{1+\frac1{\sqrt2}}}=\sqrt{(\sqrt2-1)^2}

\boxed{\tan\dfrac\pi8=\sqrt2-1}

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