I don't even know where to start.

[Karn00k]

(1+sina+cosa)/(1+sina-cosa)=ctg(a/2)

I tried pulling out sina from the numerator and denominator, but I was only left with this, which I find even more complicated:
(coseca+cota+1)/(coseca-cota+1)

I know that cot(a/2) is (1+cosa)/sina or sina/(1-cosa), but I don't even know what I could do with the right side.

 

[MarkFL]

I would start with the left side, rewrite the argument in terms of a/2 using double angle identities, and a Pythagorean identity for the 1's, to get:

\(\displaystyle \dfrac{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)-\sin^2\left(\frac{a}{2} \right)}{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)-\cos^2\left(\frac{a}{2} \right)+\sin^2\left(\frac{a}{2} \right)}\)

Now, combine terms, and factor/reduce, and you are home.

 

[Karn00k]

I would start with the left side, rewrite the argument in terms of a/2 using double angle identities, and a Pythagorean identity for the 1's, to get:

\(\displaystyle \dfrac{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)-\sin^2\left(\frac{a}{2} \right)}{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)-\cos^2\left(\frac{a}{2} \right)+\sin^2\left(\frac{a}{2} \right)}\)

Now, combine terms, and factor/reduce, and you are home.

I'm still a little confused with how you got that and after combining and reducing the terms, I got (1+cosa+(2sin(a/2))(cos(a/2)))/(1-cosa+2sin(a/2))(cos(a/2)))
What should I do next?

 

[MarkFL]

I'm still a little confused with how you got that and after combining and reducing the terms, I got (1+cosa+(2sin(a/2))(cos(a/2)))/(1-cosa+2sin(a/2))(cos(a/2)))
What should I do next?

To get the result I posted, I used the following identities:

\(\displaystyle \sin^2(\theta)+\cos^2(\theta)=1\)

\(\displaystyle \sin(2\theta)=2\sin(\theta)\cos(\theta)\)

\(\displaystyle \cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)\)

and so applying these we get:

\(\displaystyle \dfrac{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)-\sin^2\left(\frac{a}{2} \right)}{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)-\cos^2\left(\frac{a}{2} \right)+\sin^2\left(\frac{a}{2} \right)}\)

Combine like terms:

\(\displaystyle \dfrac{2\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)}{2\sin^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)}\)

Factor:

\(\displaystyle \dfrac{2\cos\left(\frac{a}{2} \right)\left(\cos\left(\frac{a}{2} \right)+\sin\left(\frac{a}{2} \right) \right)}{2\sin\left(\frac{a}{2} \right)\left(\sin\left(\frac{a}{2} \right)+\cos\left(\frac{a}{2} \right) \right)}\)

Reduce:

\(\displaystyle \dfrac{\cos\left(\frac{a}{2} \right)}{\sin\left(\frac{a}{2} \right)}\)

Rewrite using the definition of the cotangent function:

\(\displaystyle \cot\left(\frac{a}{2} \right)\)

 

[Karn00k]

To get the result I posted, I used the following identities:

\(\displaystyle \sin^2(\theta)+\cos^2(\theta)=1\)

\(\displaystyle \sin(2\theta)=2\sin(\theta)\cos(\theta)\)

\(\displaystyle \cos(2\theta)=\cos^2(\theta)-\sin^2(\theta)\)

and so applying these we get:

\(\displaystyle \dfrac{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)-\sin^2\left(\frac{a}{2} \right)}{\sin^2\left(\frac{a}{2} \right)+\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)-\cos^2\left(\frac{a}{2} \right)+\sin^2\left(\frac{a}{2} \right)}\)

Combine like terms:

\(\displaystyle \dfrac{2\cos^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)}{2\sin^2\left(\frac{a}{2} \right)+2\sin\left(\frac{a}{2} \right)\cos\left(\frac{a}{2} \right)}\)

Factor:

\(\displaystyle \dfrac{2\cos\left(\frac{a}{2} \right)\left(\cos\left(\frac{a}{2} \right)+\sin\left(\frac{a}{2} \right) \right)}{2\sin\left(\frac{a}{2} \right)\left(\sin\left(\frac{a}{2} \right)+\cos\left(\frac{a}{2} \right) \right)}\)

Reduce:

\(\displaystyle \dfrac{\cos\left(\frac{a}{2} \right)}{\sin\left(\frac{a}{2} \right)}\)

Rewrite using the definition of the cotangent function:

\(\displaystyle \cot\left(\frac{a}{2} \right)\)

thank you!