How do you use a half-angle formula to simplify $csc [ (7pi) / 8 ]$ ?

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Answer 1 · 2015-07-06T04:56:07

Find $csc ((7pi)/8)$

$csc ((7pi)/8) = 1/sin ((7pi)/8)$

Call $sin ((7pi)/8) = sin a$ (Quadrant II)

$cos 2a = cos ((14pi)/8) = cos ((pi)/4 + 2pi) = cos ((pi)/4) = (sqrt2)/2$

Use trig identity $cos 2a = 1 - 2sin^2 a.$

$(sqrt2)/2 = 1 - 2sin^2a$ --> $sin^2 a = (2 - sqrt2)/4$

$sin a = sin ((7pi)/8) = sqrt(2 + sqrt2)/2$

$csc ((7pi)/8) = 1/sin a = 2/(sqrt(2 + sqrt2)$

How do you use a half-angle formula to simplify csc [ (7pi) / 8 ]csc [ (7pi) / 8 ]?