To prove cos3theta +2cos5theta +cos7theta/costheta +2cos3theta +cos5theta=cos2theta-sin2theta×tan3theta ?

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To prove cos3theta +2cos5theta +cos7theta/costheta +2cos3theta +cos5theta=cos2theta-sin2theta×tan3theta ?

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Answer 1 · 2018-07-22T15:11:18

Please refer to a Proof in Explanation.

Explanation:

$The Expression = \frac{cos 3 θ + 2 cos 5 θ + cos 7 θ}{cos θ + 2 cos 3 θ + cos 5 θ}$ $"The Expression"=(cos3theta+2cos5theta+cos7theta)/(costheta+2cos3theta+cos5theta)$ .

$The Nr. = (cos 3 θ + 2 cos 5 θ + cos 7 θ)$ $"The Nr."=(cos3theta+2cos5theta+cos7theta)$

$= (cos 7 θ + cos 3 θ) + 2 cos 5 θ$ $=(cos7theta+cos3theta)+2cos5theta$ ,

$= 2 cos (\frac{7 θ + 3 θ}{2}) cos (\frac{7 θ - 3 θ}{2}) + 2 cos 5 θ$ $=2cos((7theta+3theta)/2)cos((7theta-3theta)/2)+2cos5theta$ ,

$= 2 cos 5 θ cos 2 θ + 2 cos 5 θ$ $=2cos5thetacos2theta+2cos5theta$ ,

$:." The Nr."=2cos5theta(cos2theta+1)$ .

$"On similar lines the Dr."=2cos3theta(cos2theta+1)$ .

$:."The Exp."=(cos5theta)/(cos3theta)$ ,

$={cos(3theta+2theta)}/(cos3theta)$ ,

$={cos3thetacos2theta-sin3thetasin2theta}/(cos3theta)$ ,

$=(cos3thetacos2theta)/(cos3theta)-(sin3thetasin2theta}/(cos3theta)$ ,

$=cos2theta-(sin2theta){(sin3theta)/(cos3theta)}$ ,

$=cos2theta-sin2theta*tan3theta$ , as desired!

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To prove cos3theta +2cos5theta +cos7theta/costheta +2cos3theta +cos5theta=cos2theta-sin2theta×tan3theta ?

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