cot(theta)+tan(theta)cot(θ)+tan(θ)
Its equal to
1/tan(theta)+tan(theta)1tan(θ)+tan(θ)
Get the lcd and add which will result to
(1+tan^2(theta))/tan(theta)1+tan2(θ)tan(θ)
One of trigonometric identities is the 1+tan^2(theta)=sec^2(theta)1+tan2(θ)=sec2(θ)
So the numerator would be replaced.
sec^2(theta)/tan(theta)sec2(θ)tan(θ)
Another trigonometric identities are the
sec(theta)=1/cos(theta)sec(θ)=1cos(θ)
tan(theta)=sin(theta)/cos(theta)tan(θ)=sin(θ)cos(θ)
so changing the numerator and denominator will result to
1/cos^2(theta)/sin(theta)/cos(theta)1cos2(θ)/sin(θ)cos(θ)
1/(cos(theta) * cancel(cos(theta)))*cancel(cos(theta))/sin(theta)
=color(red)(1/(cos(theta)sin(theta))
If you understand double angle formulae...
1/(costhetasintheta) = 1/ (1/2 * (2sinthetacostheta)) = 1/( 1/2 * sin2theta)
= 2/sin(2theta) = color(red)(2 csc2theta