Question

How do you use the half-angle identity to find the exact value of tan(7pi/8)?

Answer 1

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

Ans:`1 - sqrt2`

Explanation:

`tan ((7pi)/8) = tan (-pi/8 + pi) = tan (-pi/8) = - tan (pi/8)`
Call `tan (pi/8) = t`
`tan 2t = tan ((2pi)/8) = tan (pi/4) = 1`
Use trig identity: `tan 2t = 1 = (2t)/(1 - t^2)` -->Solve quadratic equation:

`t^2 + 2t - 1 = 0`
`D = d^2 = 4 + 4 = 8`--> `d = +- 2sqrt2`
`t = -1 +- sqrt2`. Accept only positive answer:
`t = tan (pi/8) = (sqrt2 - 1)`

Finally:`tan ((7pi)/8) = - tan (pi/8) = 1 - sqrt2`