How do you find the exact value of $cos (- \frac{5 π}{12})$ $cos(-(5pi)/12)$ using the half angle formula?

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Answer 1 · 2015-08-30T14:39:30

$color(red)(cos(-(5π)/12) =sqrt(2–sqrt3)/2)$

The cosine half-angle formula is

$cos(x/2) = ±sqrt((1 + cos x) / 2)$

The sign is positive if $x/2$ is in the first or fourth quadrant and negative if $x/2$ is in the second or third quadrant.

$-(5π)/12$ is in the fourth quadrant, so the sign is positive.

Also, $cos(-x) = cosx$ , and

$(5π)/12 = 1/2×(5π)/6$

∴ $cos(-(5π)/12) =cos((5π)/12) = cos(((5π)/6)/2) = sqrt((1+cos ((5π)/6))/2)$

$cos(-(5π)/12) =sqrt((1+cos (π - π/6))/2) == sqrt((1 – cos(π/6))/2)$

$cos(-(5π)/12) = sqrt((1 – (sqrt3)/2)/2) = sqrt((2 - sqrt3)/4)$

$cos(-(5π)/12) = sqrt(2 - sqrt3)/2$

How do you find the exact value of cos(-(5pi)/12)cos(−5π12)cos(-(5pi)/12) using the half angle formula?