How do you use product to sum formulas to write the product $cos 5 θ cos 3 θ$ $cos5thetacos3theta$ as a sum or difference?

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Answer 1 · 2017-04-25T12:57:35

$cos 5 θ cos 3 θ = \frac{1}{2} cos 8 θ + \frac{1}{2} cos 2 θ$ $cos5thetacos3theta=1/2cos8theta+1/2cos2theta$

As $cos (A + B) = cos A cos B - sin A sin B$ $cos(A+B)=cosAcosB-sinAsinB$ and

$cos (A - B) = cos A cos B + sin A sin B$ $cos(A-B)=cosAcosB+sinAsinB$

Adding two, we get $cos (A + B) + cos (A - B) = 2 cos A cos B$ $cos(A+B)+cos(A-B)=2cosAcosB$

or $cos A cos B = \frac{1}{2} [cos (A + B) + cos (A - B)]$ $cosAcosB=1/2[cos(A+B)+cos(A-B)]$

Hence, $cos 5 θ cos 3 θ$ $cos5thetacos3theta$

= $\frac{1}{2} [cos (5 θ + 3 θ) + cos (5 θ - 3 θ)]$ $1/2[cos(5theta+3theta)+cos(5theta-3theta)]$

= $\frac{1}{2} cos 8 θ + \frac{1}{2} cos 2 θ$ $1/2cos8theta+1/2cos2theta$

How do you use product to sum formulas to write the product cos5thetacos3thetacos5θcos3θcos5thetacos3theta as a sum or difference?