# How do you find the value of #cos((7pi)/8)# using the double or half angle formula?

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Using the half-angle formula for cosine, (\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}), where the sign depends on the quadrant of (\frac{\theta}{2}), you can find the value of (\cos(\frac{7\pi}{8})) as follows:

- Rewrite (\frac{7\pi}{8}) as (\frac{14\pi}{16}).
- Apply the half-angle formula: (\cos(\frac{7\pi}{16}) = \pm \sqrt{\frac{1 + \cos(\frac{14\pi}{16})}{2}}).
- Determine the sign: Since (\frac{7\pi}{16}) lies in the second quadrant where cosine is negative, the negative sign applies.
- Substitute the known values: (\cos(\frac{7\pi}{16}) = - \sqrt{\frac{1 + \cos(\frac{14\pi}{16})}{2}}).
- Calculate (\cos(\frac{14\pi}{16})). This can be simplified as (\cos(\frac{\pi}{2}) = 0).
- Substitute (\cos(\frac{14\pi}{16}) = 0) into the formula.
- Simplify to find the value of (\cos(\frac{7\pi}{8})).

So, (\cos(\frac{7\pi}{8}) = - \sqrt{\frac{1 + 0}{2}} = - \sqrt{\frac{1}{2}} = -\frac{\sqrt{2}}{2}).

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