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

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

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Answer 1 · 2016-02-18T06:46:27

$f (θ) = \frac{1}{\sqrt{\frac{1}{2} + \sqrt{\frac{1 + cos θ}{8}}}}$ $f(theta)= 1/sqrt ( 1/2 +sqrt((1+cos theta)/8$ + $\sqrt{\frac{1}{2} + \sqrt{\frac{1 + cos θ}{8}}}$ $sqrt ( 1/2 +sqrt((1+cos theta)/8$

Explanation:

= $\frac{sec θ}{4} + sin (\frac{π}{2} - \frac{θ}{4}) = \frac{1}{cos (\frac{θ}{4})} + cos (\frac{θ}{4})$ $sec theta/4 + sin(pi/2-theta/4)= 1/cos (theta/4 )+ cos (theta/4)$

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

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

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

= $\sqrt{\frac{1}{2} + \sqrt{\frac{1 + cos θ}{8}}}$ $sqrt ( 1/2 +sqrt((1+cos theta)/8$

Hence $f (θ) = \frac{1}{\sqrt{\frac{1}{2} + \sqrt{\frac{1 + cos θ}{8}}}}$ $f(theta)= 1/sqrt ( 1/2 +sqrt((1+cos theta)/8$ + $\sqrt{\frac{1}{2} + \sqrt{\frac{1 + cos θ}{8}}}$ $sqrt ( 1/2 +sqrt((1+cos theta)/8$

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

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