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Prove that: (cos6theta+6cos4theta+15 cos...

Prove that: `(cos6theta+6cos4theta+15 cos2theta+10)/(cos5theta+5cos3theta+10 costheta)=2costheta`

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`cos 6 theta+6 cos 4 theta+15 cos 2 theta+10`

`=(cos 6 theta+cos 4 theta)+5(cos 4 theta+cos 2 theta)+10(cos 2 theta+1)`

`=(2 cos (frac{6 theta+4 theta}{2}) cos (frac{6 theta-4 theta}{2}))+5(2 cos (frac{4 theta+2 theta}{2}) cos (frac{4 theta-2 theta}{2}))+` `10(2 cos ^{2} theta)`

`=2 cos 5 theta cos theta+10 cos 3 theta cos theta+20 cos ^{2} theta`

`=2(cos 5 theta+5 cos 3 theta+10 cos theta) cos theta`

`therefore frac{cos 6 theta+6 cos 4 theta+15 cos 2 theta+10}{cos 5 theta+5 cos 3 theta+10 cos theta}=2 cos theta`

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