Question

How do you simplify `f(theta)=-2csc(theta/4)+tan(theta/2)-3cos(theta/4)` to trigonometric functions of a unit `theta`?

Answer 1

`f(theta)=-2csc(theta/4)+tan(theta/2)-3cos(theta/4)`

`=frac{-2}{sin(theta/4)}+frac{sin(theta/2)}{cos(theta/2)}-3cos(theta/4)`

`=frac{-2cos(theta/2)+sin(theta/2)sin(theta/4)-3cos(theta/4)sin(theta/4)cos(theta/2)}{sin(theta/4)cos(theta/2)}`

Some identities to substitute in (half-angle and quarter-angle formulas):
`sin(a/2)=+-sqrt[frac{1-cosa}{2}]`
`cos(a/2)=+-sqrt[frac{1+cosa}{2}]`
`sin(theta/4)=sin((theta/2)/2)=+-sqrt[frac{1-+-sqrtfrac{1+costheta}{2}}{2}]`
`cos(theta/4)=cos((theta/2)/2)=+-sqrtfrac{1++-sqrtfrac{1-costheta}{2}}{2}`