Prove the following:`sin(pi/6+A).cos(pi/3-B)+sin(pi/3-B).cos(pi/6+A)=cos(A-B)`

Prove the following: cos(pi/4-x).cos(pi/4-y)-sin(pi/4-x).sin(pi/4-y)=sin(x+y)

Prove the following: cos(pi/2-x)cos(pi/2-y)-sin(pi/2-x)sin(pi/2-y)=-cos(x+y)

Prove the following: 4cosxcos(pi/3+x)cos(pi/3-x)=cos3x

Prove the following: (frac(cos(pi+x)cos(pi-x))(sin(pi-x)cos(pi/2+x)))=cot^2x

Prove the following: cos(3pi/2+x)cos(2pi+x)[cot(3pi/2-x)+cot(2pi+x)]=1

Prove the following: (frac(sin^3(pi+x)sec^2(pi-x)tan(2pi-x))(cos^2(pi/2+x)sin(pi-x)cosec^2(-x)))=tan^3x

cos(pi/5)cos((2pi)/5)cos((4pi)/5)cos((8pi)/5)=

Prove the following: (1+cos (pi/8))(1+cos((3pi)/8))(1+cos ((5pi)/8))(1+cos ((7pi)/8))=1/8

Using tables evaluate the following: cos^2 0+cos^2 (pi/6)+cos^2 (pi/3)+cos^2 (pi/2)

Use De Moivers theorem and simplify the following: frac{(cos 7(pi)/13+ i sin 7(pi)/13)^4}{(cos 4(pi)/13 - i sin 4(pi)/13)^6}