tan(theta)-cot(theta)+sin(theta)
We have to write in terms of cos(theta)
color(blue)"Let us start by using the identity"
tan(theta) = sin(theta)/cos(theta) and cot(theta) = cos(theta)/sin(theta)
We get
tan(theta)-cot(theta)+sin(theta)
=sin(theta)/cos(theta) - cos(theta)/sin(theta) + sin(theta)
color(blue)"In order to simplify we need to use Least Common Denominator for all the fractions"
=(sin(theta)sin(theta))/(cos(theta)sin(theta)) -(cos(theta)cos(theta))/(cos(theta)sin(theta)) + (sin(theta)cos(theta)sin(theta))/(cos(theta)sin(theta))
=(sin^2(theta)-cos^2(theta) + sin^2(theta)cos(theta))/(cos(theta)sin(theta))
=(1-cos^2(theta)-cos^2(theta)+(1-cos^2(theta))cos(theta))/(cos(theta)sin(theta))
=((1-2cos^2(theta))+cos(theta)-cos^3(theta))/(cos(theta)sin(theta))
=(-cos^3(theta)-2cos^2(theta)+cos(theta)+1)/(cos(theta)sin(theta))
=(-cos^3(theta)-2cos^2(theta)+cos(theta)+1)/(cos(theta)sqrt(1-cos^2theta))