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

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Answer 1 · 2016-01-26T09:05:52

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

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..

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