We shall use the following identities:
#sectheta= 1/costheta#
#cottheta= costheta/sintheta#
Also, #cos^2theta+sin^2theta=1#
#=>cos^2theta=1-sin^2theta color(red)rarr costheta=+-sqrt(1-sin^2theta)#
The #+# or #-# depends on the quadrant in which the angle #theta#
is found.
Assuming #theta# acute we can simply ignore this.
Hence,
#costheta= sqrt(1-sin^2theta)#
#=>1/costheta=1/(sqrt(1-sin^2theta))#
#costheta/sintheta=sqrt(1-sin^2theta)/sintheta#
#=>4costheta-sectheta+2cottheta color(red)rarr 4*(sqrt(1-sin^2theta))-(1/(sqrt(1-sin^2theta)))+2*(sqrt(1-sin^2theta)/sintheta)#
#color(red)rarr [4(1-sin^2theta)sintheta-sintheta+2(1-sin^2theta)]/(sinthetasqrt(1-sin^2theta))#
