Question

Prove that ----1/2 (cos :2 theta-cos 8theta)=sin 5theta.sin 3theta​

Answer 1

Prove that:

$cos(2\alpha )-cos(8\alpha )=sin(5\alpha )sin(3\alpha )$

Remember that:

$cos(A+B)=cos(A)cos(B)-sin(A)sin(B)\\cos(A-B)=cos(A)cos(B)+sin(A)sin(B)\\\\\\cos(A+B)-cos(A-B)=-2sin(A)sin(B)$

We wish to remove this add (A+B).

$\left \{ {{A+B=p} \atop {A-B=q}} \right. \\\\2A=p+q$

$A=\frac{p+q}{2} \\\\B=\frac{p-q}{2}$

Then we de identity:

$cos(p)-cos(q)=-2sin(\frac{p+q}{2}) sin(\frac{p-q}{2})$

Let p=2α and q=8α

$cos(2\alpha )-cos(8\alpha )=-2sin(\frac{2\alpha +8\alpha }{2})sin(\frac{2\alpha -8\alpha }{2}) \\\\cos(2\alpha )-cos(8\alpha )=-2sin(5\alpha )sin(-3\alpha )\\\\*sin(-\alpha )=-sin(\alpha )\\\\cos(2\alpha )-cos(8\alpha )=2sin(5\alpha )sin(3\alpha )$