How do you express cos( (4 pi)/3 ) * cos (( pi) /4 ) cos(4π3)cos(π4) without using products of trigonometric functions?

1 Answer

I suggest , use "Sum of Trigonometric Functions"
cos ((4pi)/3)* cos(pi/4)=1/2*cos ((19pi)/12)+1/2*cos ((13pi)/12)cos(4π3)cos(π4)=12cos(19π12)+12cos(13π12)

Explanation:

Use Double-Angle Formulas

cos (A+B)=cos A cos B-sin A sin Bcos(A+B)=cosAcosBsinAsinB

cos (A-B)=cos A cos B+sin A sin Bcos(AB)=cosAcosB+sinAsinB

Add the left side terms equals the sum of the right terms:

cos (A+B)+cos (A-B)=cos(A+B)+cos(AB)=
cos A cos B-cancel(sin A sin B)+cos A cos B+cancel(sin A sin B)

cos (A+B)+cos (A-B)=2*cos A cos B

it follows

cos A cos B=1/2*cos (A+B)+1/2*cos (A-B)

Use the given: Let A=(4pi)/3 and B=pi/4

it follows

cos ((4pi)/3) cos (pi/4)=

1/2*cos ((4pi)/3+pi/4)+1/2*cos ((4pi)/3-pi/4)

Finally, after simplification

cos ((4pi)/3)* cos(pi/4)=1/2*cos ((19pi)/12)+1/2*cos ((13pi)/12)

Have a nice day !!! from the Philippines..