Use a sum or difference identity to find the exact value of

Use a sum or difference identity to find the exact value of the expression.

tan 15 degrees=

(simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution

tan(15)

Use the formula: tanx = sinx/cosx

Now use half angle formula:

sinx = sqrt[(1 -cos2x)/2]

sin15 = sqrt[ (1- cos30)/2]

= sqrt[ (2-sqrt3)]/2

cosx = sqrt[(1+cos2x)/2]

cos15 = sqrt[(1+cos30)/2]

= sqrt[( 1+sqrt3/2)/2]

= sqrt[2+sqrt3]/2

So, tan15 = sin15/cos15 = sqrt[(2- sqrt3)/(2 +sqrt3)]