sec(theta) + cos^2(theta) - cos(theta)sec(θ)+cos2(θ)−cos(θ)
To simplify in terms of sin(theta)sin(θ) let us write sec(theta)sec(θ) as 1/cos(theta)1cos(θ)
=1/cos(theta) + cos^2(theta) - cos(theta)=1cos(θ)+cos2(θ)−cos(θ)
=1/cos(theta) + (cos^2(theta)cos(theta))/cos(theta) - (cos(theta)cos(theta))/cos(theta)=1cos(θ)+cos2(θ)cos(θ)cos(θ)−cos(θ)cos(θ)cos(θ)
= (1+cos^3(theta)-cos^2(theta))/cos(theta)=1+cos3(θ)−cos2(θ)cos(θ)
=(1-cos^2(theta) + cos(theta)cos^2(theta))/cos(theta)=1−cos2(θ)+cos(θ)cos2(θ)cos(θ)
=(sin^2(theta)+sqrt(1-sin^2(theta))(1-sin^2(theta)))/sqrt(1-sin^2(theta)=sin2(θ)+√1−sin2(θ)(1−sin2(θ))√1−sin2(θ)
If you need it can be simplified further as
=sin^2(theta)/sqrt(1-sin^2(theta)) + (sqrt(1-sin^2(theta))(1-sin^2(theta)))/sqrt(1-sin^2(theta)=sin2(θ)√1−sin2(θ)+√1−sin2(θ)(1−sin2(θ))√1−sin2(θ)
=sin^2(theta)/sqrt(1-sin^2(theta)) + (cancel(sqrt(1-sin^2(theta)))(1-sin^2(theta)))/cancel(sqrt(1-sin^2(theta))
=sin^2(theta)/sqrt(1-sin^2(theta)) + 1- sin^2(theta)
