Question

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

Answer 1

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

Explanation:

Use Double-Angle Formulas

`cos (A+B)=cos A cos B-sin A sin B`

`cos (A-B)=cos A cos B+sin A sin B`

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

`cos (A+B)+cos (A-B)=`
`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..