How do you express cos( (15 pi)/ 8 ) * cos (( pi) /3 ) cos(15π8)cos(π3) without using products of trigonometric functions?

1 Answer

It is cos(15pi/8)*cos(pi/3)=1/2[cos(5*pi/24)+cos(37/24*pi)]cos(15π8)cos(π3)=12[cos(5π24)+cos(3724π)]

Explanation:

Well we know that

cos(A+B)=cosA*cosB-sinA*sinBcos(A+B)=cosAcosBsinAsinB

and

cos(A-B)=cosA*cosB+sinA*sinBcos(AB)=cosAcosB+sinAsinB

Add these two to get

cosA*cosB=1/2[cos(A+B)+cos(A-B)]cosAcosB=12[cos(A+B)+cos(AB)]

Hence for A=15pi/8A=15π8 and B=pi/3B=π3 we have that

cos(15pi/8)*cos(pi/3)=1/2[cos(15pi/8+pi/3)+cos(15pi/8-pi/3)]=> cos(15pi/8)*cos(pi/3)=1/2[cos(53/24*pi)+cos(37/24*pi)]=> cos(15pi/8)*cos(pi/3)=1/2[cos(5*pi/24)+cos(37/24*pi)]