Find the exact value by using a halfangle identity Tan7 pi8

Find the exact value by using a half-angle identity. Tan[7 pi/8] -1 + squareroot 2 1 - squareroot 2 -1 - squareroot 2 must show work correctly to recieve any credit [+2]

Solution

tan (7/8) = tan (7/4 /2). So we use the half angle formula for tangent, which is:

tan A/2 = sin A / (1+cosA)

tan (7/4 /2) = (sin 7/4) / (1+cos 7/4)
= -2/2 / (1+2/2)
= -2 / (2 + 2)
= -2 (2-2) / 2
= (2 - 22) / 2
= 1 - 2