How do you simplify $f (θ) = - 2 csc (\frac{θ}{4}) - cot (\frac{θ}{2}) + 2 sin (\frac{θ}{4})$ $f(theta)=-2csc(theta/4)-cot(theta/2)+2sin(theta/4)$ to trigonometric functions of a unit $θ$ $theta$ ?

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How do you simplify $f (θ) = - 2 csc (\frac{θ}{4}) - cot (\frac{θ}{2}) + 2 sin (\frac{θ}{4})$ $f(theta)=-2csc(theta/4)-cot(theta/2)+2sin(theta/4)$ to trigonometric functions of a unit $θ$ $theta$ ?

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Answer 1 · 2016-07-23T00:32:51

$= \frac{- 2 (1 + \sqrt{\frac{1}{2} (1 + cos θ)})}{\sqrt{\frac{1}{2} (1 - \sqrt{\frac{1}{2} (1 + cos θ)})}} - \frac{1 + cos θ}{sin θ}$ $=(-2(1+sqrt(1/2(1+costheta))))/sqrt(1/2(1-sqrt(1/2(1+costheta))))-(1+costheta)/sintheta$

Explanation:

$f (θ) = - 2 csc (\frac{θ}{4}) - cot (\frac{θ}{2}) + 2 sin (\frac{θ}{4})$ $f(theta)=-2csc(theta/4)-cot(theta/2)+2sin(theta/4)$

$= 2 sin (\frac{θ}{4}) - \frac{2}{sin (\frac{θ}{4})} - cot (\frac{θ}{2})$ $=2sin(theta/4)-2/sin(theta/4)-cot(theta/2)$

$= \frac{2 {sin}^{2} (\frac{θ}{4}) - 2}{sin (\frac{θ}{4})} - \frac{2 {cos}^{2} (\frac{θ}{2})}{2 sin (\frac{θ}{2}) cos (\frac{θ}{2})}$ $=(2sin^2(theta/4)-2)/sin(theta/4)-(2cos^2(theta/2))/(2sin(theta/2)cos(theta/2))$

$= \frac{- 2 (1 - {sin}^{2} (\frac{θ}{4}))}{sin (\frac{θ}{4})} - \frac{1 + cos θ}{sin θ}$ $=(-2(1-sin^2(theta/4)))/sin(theta/4)-(1+costheta)/sintheta$

$= \frac{- 2 {cos}^{2} (\frac{θ}{4})}{sin (\frac{θ}{4})} - \frac{1 + cos θ}{sin θ}$ $=(-2cos^2(theta/4))/sin(theta/4)-(1+costheta)/sintheta$

$= \frac{- 2 (1 + cos (\frac{θ}{2}))}{\sqrt{\frac{1}{2} (1 - cos (\frac{θ}{2}))}} - \frac{1 + cos θ}{sin θ}$ $=(-2(1+cos(theta/2)))/sqrt(1/2(1-cos(theta/2)))-(1+costheta)/sintheta$

$= \frac{- 2 (1 + \sqrt{\frac{1}{2} (1 + cos θ)})}{\sqrt{\frac{1}{2} (1 - cos (\frac{θ}{2}))}} - \frac{1 + cos θ}{sin θ}$ $=(-2(1+sqrt(1/2(1+costheta))))/sqrt(1/2(1-cos(theta/2)))-(1+costheta)/sintheta$

$= \frac{- 2 (1 + \sqrt{\frac{1}{2} (1 + cos θ)})}{\sqrt{\frac{1}{2} (1 - \sqrt{\frac{1}{2} (1 + cos θ)})}} - \frac{1 + cos θ}{sin θ}$ $=(-2(1+sqrt(1/2(1+costheta))))/sqrt(1/2(1-sqrt(1/2(1+costheta))))-(1+costheta)/sintheta$

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How do you simplify $f (θ) = - 2 csc (\frac{θ}{4}) - cot (\frac{θ}{2}) + 2 sin (\frac{θ}{4})$ $f(theta)=-2csc(theta/4)-cot(theta/2)+2sin(theta/4)$ to trigonometric functions of a unit $θ$ $theta$ ?

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How do you simplify $f (θ) = - 2 csc (\frac{θ}{4}) - cot (\frac{θ}{2}) + 2 sin (\frac{θ}{4})$ $f(theta)=-2csc(theta/4)-cot(theta/2)+2sin(theta/4)$ to trigonometric functions of a unit $θ$ $theta$ ?