asin x using the difference identity bcos x using the halfan

a)sin x, using the difference identity

b)cos x, using the half-angle identity

Solution

x= pi/12

a) sinx = sinpi/12

= sin( pi/3 - pi/4)

Use sin(x-y) = sinxcosy - cosxsiny

sinpi/3cospi/4 - cospi/3sinpi/4

sqrt(3)/2*1/sqrt2 - (1/2)*(1/sqrt2)

[ sqrt(3) - 1]/2sqrt(2)

b) cosx = cos(pi/12)

cos((pi/6)/2)

cos(x/2) = sqrt[(1+cosx)/2]

cos(pi/12) = sqrt[( 1+sqrt(3)/2)/2]

= sqrt[(2+sqrt3)]/2