Given: #csc(x) = 8, 90^@ < x < 180^@#
Use the identity #csc(x) = 1/sin(x)#
#sin(x) = 1/8, 90^@ < x < 180^@#
Use the identity:
#cos(x) = +-sqrt(1-sin^2(x))#
Because of we are told that #x# is in the second quadrant we choose the negative value:
#cos(x) = -sqrt(1-sin^2(x)), 90^@ < x < 180^@#
#cos(x) = -sqrt(1-(1/8)^2), 90^@ < x < 180^@#
#cos(x) = -sqrt(64/64-1/64), 90^@ < x < 180^@#
#cos(x) = -sqrt(63/64), 90^@ < x < 180^@#
#cos(x) = -sqrt63/8, 90^@ < x < 180^@#
#cos(x) = -3sqrt7/8, 90^@ < x < 180^@#
Use the identity:
#sin(x/2) = +-sqrt((1-cos(x))/2)#
The domain of #x/2# is #45^@ < x/2 < 90^@#, therefore, we choose the positive value:
#sin(x/2) = sqrt((1-(-3sqrt7/8))/2)#
#sin(x/2) = sqrt((1/2+3/16sqrt7)#
Use the identity:
#cos(x/2) = +-sqrt((1+cos(x))/2)#
The domain of #x/2# is #45^@ < x/2 < 90^@#, therefore, we choose the positive value:
#cos(x/2) = sqrt((1/2-3/16sqrt7)#
Use the identity
#tan(x/2) = sin(x)/(1-cos(x))#
#tan(x/2) = (1/8)/(1+3/8sqrt7)#
#tan(x/2) = 1/(8+3sqrt7)#
#tan(x/2) = 8-3sqrt7#
