Question

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

Answer 1

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)`