"The Expression"=(cos3theta+2cos5theta+cos7theta)/(costheta+2cos3theta+cos5theta).
"The Nr."=(cos3theta+2cos5theta+cos7theta)
=(cos7theta+cos3theta)+2cos5theta,
=2cos((7theta+3theta)/2)cos((7theta-3theta)/2)+2cos5theta,
=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!