Question

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

Answer 1

`f(theta)= 1/sqrt ( 1/2 +sqrt((1+cos theta)/8` + `sqrt ( 1/2 +sqrt((1+cos theta)/8`

Explanation:

=`sec theta/4 + sin(pi/2-theta/4)= 1/cos (theta/4 )+ cos (theta/4)`

`cos (theta/4)= sqrt(cos^2(theta/4))=sqrt(1/2 (1+cos(theta/2)`

=`sqrt(1/2 (1+sqrt (cos^2( theta/2)`

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

=`sqrt ( 1/2 +sqrt((1+cos theta)/8`

Hence `f(theta)= 1/sqrt ( 1/2 +sqrt((1+cos theta)/8` + `sqrt ( 1/2 +sqrt((1+cos theta)/8`