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

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Answer 1 · 2015-05-22T16:34:34

Use the trig identity: $tan^2 x = (1 - cos 2x)/(1 + cos 2x)$

$tan^2 ((7pi)/8) = (1 - cos (7pi)/4)/(1 + cos (7pi)/4$ =

or $cos ((7pi)/4)= cos (pi/4) = 1/sqrt2$ , then:

$tan^2 (7pi/8) = (sqrt2 - 1)/(sqrt2+ 1)$ =

$= (sqrt2 - 1)^2/(2 - 1) = (sqrt2 - 1)^2$

Finally, $tan ((7pi)/8) = (sqrt2 - 1)$

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