Maths M2

1a)
cos(5π/12)
=cos(π/6+π/4)
=cos(π/6)cos(π/4)-sin(π/6)sin(π/4)
=√(3)/2*√(2)/2-1/2*√(2)/2
=√(6)/4-√(2)/4
=(√6-√4)/4

cos(5π/12)=(√6-√4)/4

1b)
tan(π/6)=tan[2(π/12)]
1/√(3)=2tan(π/12)/[1-tan(π/12)^2]
1-tan(π/12)^2=2√(3)tan(π/12)
∴tan(π/12)^2+2√(3)tan(π/12)-1=0
tan(π/12)= 2-√(3) or -2-√(3) (rejected) [∵tan(π/12) is in quadrant I]

tan(π/12)= 2-√(3)

2)
tan(A+B)=1/2
1/2=(tanA+tanB)/(1-tanAtanB)
Sub tanA=2,
1/2=(2+tanB)/(1-2tanB)
1-2tanB=4+2tanB
4tanB= -3
tanB= -3/4

3)
cos(A-B)sin(A+B)
=(cosAcosB+sinAsinB)(sinAcosB+cosAsinB)
=sinAcosAcos(B)^2+cos(A)^2sinBcosB+sin(A)^2sinBcosB+sinAcosAsin(B)^2
=sinAcosA[sin(B)^2+cos(B)^2]+sinBcosB[sin(A)^2+cos(A)^2]
=sinAcosA+sinBcosB

Formula for reference:
sin (A± B) = sinAcosB ± cos Asin B
cos(A± B) = cosAcosB (-/+) sin Asin B
tan (A±B) = (tanA ± tanB)/[1(-/+) tanAtanB]