Proving this trig identity:$\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$ 
$$\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$$

What I've tried,
$$\frac{((1+\cos\theta)+(\sin\theta))((1+\cos\theta)+(\sin\theta))}{(1+\cos\theta-\sin\theta) (1+\cos\theta+\sin\theta)}$$
$$=\frac{(1+\cos\theta+\sin\theta)^2}{(1+\cos\theta)^2-\sin^2\theta}$$
After simplifying,
$$=\frac{2(1+\sin\theta \cos\theta + \sin\theta + \cos\theta)}{\cos\theta(\cos\theta+2)}$$
I cant carry on further. Thus, any kind assistance would be much appreciated.
 A: Multiplying and dividing by $(1+\sin\theta -\cos \theta)$ yields
$$\frac{(1+\sin \theta)^2-\cos^2 \theta}{1-\cos^2\theta -\sin^2\theta +2\sin\theta \cos \theta}= \frac{1+\sin^2 \theta + 2\sin \theta -\cos^2 \theta}{2\sin \theta \cos \theta}$$
Substituting $-\cos^2 \theta = \sin^2 \theta -1$, we obtain
$$ \frac{1+\sin^2 \theta +2\sin \theta +\sin^2 \theta -1}{2 \sin \theta \cos \theta}= \frac{2 \sin \theta(1+\sin \theta)}{2 \sin \theta \cos \theta}$$
$$=\color{red} {\frac{1+\sin \theta}{\cos \theta}}$$
Edit: clarifications for OP.
Consider the initial expression multiplied and divided by $(1+\sin\theta -\cos \theta)$:
$$\frac{((1+\cos \theta+\sin \theta)(1+\sin \theta - \cos \theta)}{(1+\cos \theta - \sin \theta)(1+ \sin \theta - \cos \theta)}$$
Considering the terms grouped this way and using the fact that $(a+b)(a-b)=a^2-b^2$ we can conclude that
$$\frac{((1+\sin \theta)+\cos \theta)((1+\sin \theta) - \cos \theta)}{(1+(\cos \theta - \sin \theta))(1-(\cos \theta - \sin \theta)}=\frac{(1+\sin \theta)^2-\cos^2\theta}{1-(\cos \theta - \sin \theta)^2}$$
Expanding the square in the denominator leads to the LHS of the first equality.
A: from Second last line, $$\displaystyle \frac{(1+\sin \theta+\cos \theta)^2}{(1+\cos \theta)^2-\sin^2 \theta}=\frac{2\left[1+\sin \theta  +\sin \theta\cdot \cos \theta+\sin \theta\right]}{2\cos \theta \cdot (1+\cos \theta)} = \frac{2(1+\cos \theta)(1+\sin \theta)}{2\cos \theta\cdot (1+\cos \theta)}$$
So we get $$\displaystyle \frac{1+\sin \theta}{\cos \theta}$$
Here I have solved Using double angle formula.
Given $\displaystyle \bf{L.H.S}$ as $\displaystyle \frac{1+\cos \theta+\sin \theta}{1+\cos \theta-\sin \theta}$
Let $\theta=2\phi\;,$ Then $$\displaystyle \frac{1+\cos 2\phi+\sin 2\phi }{1+\cos 2\phi-\sin 2\phi} = \frac{2\cos^2 \phi+2\sin \phi\cdot \cos \phi}{2\cos^2 \phi-2\sin \phi\cdot \cos \phi}$$
Above we use the formula $$\bullet \; 1+\cos 2\phi = 2\cos^2 \phi$$
 and $$\bullet\; 1-\cos 2\phi = 2\sin^2 \phi$$
and $$\bullet \; \sin 2\phi = 2\sin \phi\cdot \cos \phi$$ and $$\bullet\; \cos^2\phi-\sin^2 \phi = \cos 2\phi$$
So we get $$\displaystyle \frac{\cos \phi+\sin \phi}{\cos \phi-\sin \phi} = \frac{\cos \phi+\sin \phi}{\cos \phi-\sin \phi}\times \frac{\cos \phi+\sin \phi}{\cos \phi+\sin \phi} = \frac{\sin^2 \phi +\cos^2 \phi+\sin 2\phi}{\cos^2 \phi-\sin^2 \phi}$$
So we get $$\displaystyle \frac{1+\sin 2\phi}{\cos 2\phi}$$
Now Put $2\phi = \theta\;,$ We get 
$$\displaystyle  = \frac{1+\sin \theta}{\cos \theta}$$
A: The denominator should be $2\cos\theta(1+\cos\theta)$
The numerator $=2(1+\sin\theta)(1+\cos\theta)$
A: A more advanced approach (if you want to learn more):
If you want to learn a more systematic approach to solve these problems you can read about Fourier Transforms which basically is about writing a function as sum of sines and cosines $$f(t) = \sum_{\forall k} a_k\cos(k\omega t) + b_k\sin(k\omega t)$$
Rewrite the equation so you get products instead of quotients, then these Fourier transforms have special rules which let you convolve them which automatically give you trigonometric one $$\sin^2(x) + \cos^2(x) = 1$$
double angle formulas like $$2\sin(x)\cos(x) = \sin(2x)$$ 
among many others. The functions will be very simple to convolve since their fourier transforms are very simple functions.
$$\phantom{\mathcal{F}(lhs) = \mathcal{F}(rhs)}$$
A: You can try showing
$$
\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}
  -\frac{1+\sin\theta}{\cos\theta}=0
$$
that is,
$$
(1+\cos\theta+\sin\theta)\cos\theta=
(1+\cos\theta-\sin\theta)(1+\sin\theta)
$$
Right-hand side:
$$\def\ct#1{\cos^{#1}\theta}\def\st#1{\sin^{#1}\theta}
(1-\st{})(1+\st{})+\ct{}(1+\st{})=
1-\st{2}+\ct{}(1+\st{})\\
=\ct{2}+\ct{}(1+\st{})=\ct{}(\ct{}+1+\st{})
$$
