Problem with Trig Identity

Start by replacing tan and cot with their sin and cos equivalents.  
This should be  a four line proof.

So  1+sin/cos would = cos * sin/cos

>>  So  1+sin/cos would = cos * sin/cos

That is not true, nor one of the steps.

You have to methodically transform the Left-Hand-Side of the equation until it matches the right.

Half of the first step is to replace   1 + tan    with    1 + sin/cos

Woops, that part I knew.

I mean to say that

(cos/1)/(1+(sin/cos)) where the 1's and the cos cancel out left with sinx for the first part, right?

After the first step, you will still have two terms on the LHS

One of them will be     cos / (1 + sin/cos)       The other will be similar.
But there is nothing to cancel out yet.
The two terms have to simplified [ Step 2]
Then combined [ Step 3]
At this point, you will be able to factor and cancel [ Step 4]

Not sure how to simplify them.

The problem with this term   cos / (1 + sin/cos)   is that there is a fraction in the denominator.
How do you simplify that?    

If the term were   3 / (1 + 2/7)  what would you do?

I don't imagine that this identity is particularly useful, but it is quite simple.

The only trig you need to know is that   tan = sin/cos   and   cot = cos/sin
The rest is simple algebra.

The way to solve this is to assume that it is solvable and plug away.  It is really a 60 second problem.

I know what tan and cot equal. I just don't know what to do with the (1+sin/cos)