Use a halfangle formula to find the exact value of the follo

Use a half-angle formula to find the exact value of the following expression. Tan 105 degree determine an appropriate half-angle formula for sine and the measure of the angle. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal.) tan 105 degree = 1 + sing degree/cos degree tan 105 degree = 1 - sin degree/cos degree tan 105 degree = 1 + cos degree/sin degree tan 105 degree = 1 - cos degree/sin degree tan 105 degree = (Simplify your answer, including any radicals. Use integers of fractions for any numbers in the expression. Rationalize the denominator.)

Solution

tan(105)

=>tan(90+15)

=>-cot(15)

=> - cot(30/2)

now cot(x/2) = (1 + cosx)/sinx

so cot(15) = [ 1 + cos(30)] / sin(30)

cot(15) = [1 + sqrt(3)/2] / (1/2)

cot(15) = [2 + sqrt(3)]

so tan(105) = - cot(15)

=> - [ 2 + sqrt(3)]