Question

How do you simplify
\[\frac{ \sin \theta }{ 1+\cos \theta }+\frac{ 1+\cos \theta }{ \sin \theta }\]

Answer 1

First we can try to write it as one fraction, using\[\frac{ a }{ b }+\frac{ c }{ d }=\frac{ ad+bc }{ bd }\]Applied to your fractions this would give:\[\frac{ \sin^2\theta+(1+\cos \theta)^2 }{ (1+\cos \theta)\sin \theta }\]As a next step, why not work out the backets in the numerator:\[\frac{ \sin^2\theta+1+2\cos \theta + \cos^2\theta }{ (1+\cos \theta)\sin \theta }\]And yes, now we can simplify a little already, because we recognize the form sin²θ+cos²θ=1:\[\frac{ 2+2\cos \theta }{ (1+\cos \theta)\sin \theta }\]
I think you can see there is still more possible! (hint: factor out the 2 in the numerator...)

Answer 2

Thanks. I think I got it with your help!

Answer 3

YW!