How do you simplify $cos 4 θ - 3 tan 2 θ + sin 2 θ$ $cos4theta-3tan2theta+sin2theta$ to trigonometric functions of a unit $θ$ $theta$ ?

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Answer 1 · 2016-07-21T15:15:12

$cos 4 θ - 3 tan 2 θ + sin 2 θ$ $cos4theta-3tan2theta+sin2theta$

$= 1 - 2 {sin}^{2} 2 θ - 3 tan 2 θ + sin 2 θ$ $=1-2sin^2 2theta-3tan2theta+sin2theta$

$= 1 - 2 {sin}^{2} 2 θ + sin 2 θ - 3 tan 2 θ$ $=1-2sin^2 2theta +sin2theta-3tan2theta$

$= 1 + 2 sin 2 θ - sin 2 θ - 2 {sin}^{2} 2 θ - 3 tan 2 θ$ $=1+2sin2theta-sin2theta-2sin^2 2theta-3tan2theta$

$= (1 + 2 sin 2 θ) - sin 2 θ (1 + 2 sin 2 θ) - 3 tan 2 θ$ $=(1+2sin2theta)-sin2theta(1+2sin2theta)-3tan2theta$

$= (1 + 2 sin 2 θ) (1 - sin 2 θ) - 3 tan 2 θ$ $=(1+2sin2theta)(1-sin2theta)-3tan2theta$

$= (1 + \frac{4 x}{1 + x^{2}}) (1 - \frac{2 x}{2 + x^{2}}) - \frac{6 x}{1 - x^{2}}$ $=(1+(4x)/(1+x^2))(1-(2x)/(2+x^2))-(6x)/(1-x^2)$

$where x = tan θ$ $color(red)("where " x=tantheta)$

How do you simplify cos4theta-3tan2theta+sin2thetacos4θ−3tan2θ+sin2θcos4theta-3tan2theta+sin2theta to trigonometric functions of a unit thetaθtheta?