Question

How do you simplify `f(theta)=-cot(theta/2)+2sec(theta/4+pi/2)` to trigonometric functions of a unit `theta`?

Answer 1

`f(theta)=(1+costheta+2sqrt(2(1+costheta)))/sintheta`

Explanation:

`f(theta)=-cot(theta/2)+2sec(theta/4+pi/2)`

and as `sec(pi/2+A)=-cscA`

`f(theta)=-cot(theta/2)-2csc(theta/4)`

= `-(cos(theta/2)+2)/sin(theta/2)`

As `costheta=2cos^2(theta/2)-1=1-2sin^2(theta/2)`

`cos(theta/2)=sqrt((1+costheta)/2)` and

`sin(theta/2)=sqrt((1-costheta)/2)`

and `f(theta)=-(sqrt((1+costheta)/2)+2)/sqrt((1-costheta)/2)`

= `-(sqrt(1+costheta)+2sqrt2)/sqrt(1-costheta)`

multiplying numerator and denominator by `sqrt(1+costheta)`, we get

`f(theta)=-(sqrt(1+costheta)+2sqrt2)/sqrt(1-costheta)xxsqrt(1+costheta)/sqrt(1+costheta)`

= `(1+costheta+2sqrt(2(1+costheta)))/sintheta`