Half angle tan 5pi/8

The half-angle formula for tangent is:

tan (a/2) = (sin a / (1 + cos a)) = ((1 - cos a) / sin a)

Now we can plug in values:

tan (5π/8) = (sin (5π/4) / (1 + cos (5π/4)) = ((1 - cos (5π/4)) / sin (5π/4)

tan (5π/8) = (-√2/2) / (1 + (-√2/2)) = (1 - (-√2/2)) / (-√2/2)

tan (5π/8) = ((-√2/2)) / ((2 - √2) / 2) = ((2 + √2) / 2) / (-√2/2)

Now we can solve the first half:

(-√2/2) (2 / (2 - √2))

(-√2/2) ((4 + 2√2) / 2)

(-√2/2) (2 + √2)

(-2√2 - 2) / 2

-√2 - 1

tan (5pi/8) = - √2 - 1