Question

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

Answer 1

Simplify `tan ((7pi)/8)`

Explanation:

Call `tan ((7pi)/8) = t`
Use trig identity: `tan 2a = (2tan a)/(1 - tan^2 a)`

`tan ((14pi)/8) = (2t)/(1 - t^2)`

Since `tan ((14pi)/8) = tan ((6pi)/8 + pi) = tan ((3pi)/4 + pi)`=

`= - tan ((3pi)/4) = -1`

Therefor, `(2t)/(1 - t^2) = -1` -> `t^2 - 2t - 1 = 0`
Solve this quadratic equation.
`D = d^2 = b^2 - 4ac = 4 + 4 = 8 -> d = +- 2sqrt2`
`t = 1 +- sqrt2`

`t = tan ((7pi)/8) = 1 - sqrt2` (Quadrant II)