How did this sin(𝜃) /1 − cos(𝜃) end up with csc(𝜃) + cot(𝜃)

Question

My professor is a bit bad at explaining so I am holding hope for the explanation for this. sin(𝜃) /1 − cos(𝜃)= csc(𝜃) + cot(𝜃)

Answer 1

sin(θ) (1 + cos(θ))

--------------- × ---------------------

(1 - cos(θ)) (1 + cos(θ))

sin(θ) + sin(θ)cos(θ)

= ---------------------------------

1 - cos²(θ)

sin(θ) + sin(θ)cos(θ)

= ----------------------------------

sin²(θ)

sin(θ) sin(θ)cos(θ)

= ----------- + --------------------

sin²(θ) sin²(θ)

1. cos(θ)

= ----------- + -----------

sin(θ) sin(θ)

= csc(θ) + cot(θ)

Any questions?

Answer 2

Try pasting this:

\frac{\sin\theta} {1 − \cos\theta}&= \csc\theta + \cot\theta\\

\frac{\sin\theta(1+\cos\theta)} {(1 − \cos\theta)(1+\cos\theta)}&\\

\frac{\sin\theta(1+\cos\theta)} {1 − \cos^2\theta}&\\

\frac{\sin\theta(1+\cos\theta)} {\sin^2\theta}&\\

\frac{1+\cos\theta} {\sin\theta}&\\

\frac{1} {\sin\theta}+\frac{\cos\theta} {\sin\theta}&\\

\csc\theta + \cot\theta&\\

LHS&=RHS

into here:

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