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

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

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Answer 1 · 2018-07-24T10:20:20

$f (θ) = (2 \cdot \pm \sqrt{\frac{1 + cos θ}{1 - cos θ}}) - \pm \frac{1}{\sqrt{\frac{1}{2} (1 - \pm \sqrt{\frac{1}{2} (1 - cos θ)})}} - \pm \sqrt{\frac{1}{2} (1 \pm \sqrt{\frac{1}{2} (1 + cos θ)})}$ $color(indigo)(f(theta) = (2 * pm sqrt((1 + cos theta) / (1 - cos theta))) - pm 1 / (sqrt(1/2 (1 - pm sqrt(1/2 (1 - cos theta))))) - pm sqrt(1/2 (1 pm sqrt(1/2 (1 + cos theta)))$

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![https://www.onlinemathlearning.com/http://double-angle-formula.html](https://useruploads.socratic.org/A0bL1QNkSTKHmUl1Jp77_half%20angle.png)

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

$cot (\frac{θ}{2}) = \frac{1}{tan (\frac{θ}{2})} = \pm \sqrt{\frac{1 + cos θ}{1 - cos θ}}$ $color(brown)(cot (theta/2) = 1 / tan (theta/2) = pm sqrt((1 + cos theta) / (1 - cos theta))$

$sin (\frac{θ}{4}) = \pm \sqrt{\frac{1}{2} (1 - sin (\frac{θ}{2}))}$ $sin (theta/4) = pm sqrt(1 /2 (1 - sin (theta/2)))$

$csc (\frac{θ}{4}) = \frac{1}{sin (\frac{θ}{4})} = \pm \frac{1}{\sqrt{\frac{1}{2} (1 - \pm \sqrt{\frac{1}{2} (1 - cos θ)})}}$ $color(brown)(csc (theta/4) = 1 / sin (theta / 4) = pm 1 / (sqrt(1/2 (1 - pm sqrt(1/2 (1 - cos theta)))$

$cos (\frac{θ}{4}) = \pm \sqrt{\frac{1}{2} (1 + cos (\frac{θ}{2}))}$ $cos (theta/4) = pm sqrt(1 /2 (1 + cos (theta/2)))$

$cos (\frac{θ}{4}) = \pm \sqrt{\frac{1}{2} (1 \pm \sqrt{\frac{1}{2} (1 + cos θ)})}$ $color(brown)(cos (theta / 4) = pm sqrt(1/2 (1 pm sqrt(1/2 (1 + cos theta)))$

$f (θ) = (2 \cdot \pm \sqrt{\frac{1 + cos θ}{1 - cos θ}}) - \pm \frac{1}{\sqrt{\frac{1}{2} (1 - \pm \sqrt{\frac{1}{2} (1 - cos θ)})}} - \pm \sqrt{\frac{1}{2} (1 \pm \sqrt{\frac{1}{2} (1 + cos θ)})}$ $color(indigo)(f(theta) = (2 * pm sqrt((1 + cos theta) / (1 - cos theta))) - pm 1 / (sqrt(1/2 (1 - pm sqrt(1/2 (1 - cos theta))))) - pm sqrt(1/2 (1 pm sqrt(1/2 (1 + cos theta)))$

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

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