Remember the Golden Rule of Algebra:
Do unto one side as you do unto the other.
This applies even when proving identities. All of your previous steps did not change the value of the left side. When you multiplied it by cos, you did change the value and so you have to do the same to the left.
You are correct in a sense. It is a convention in mathematics that when you are proving an identity, you generally rearrange one side while ignoring the other.
Notice the word rearrange! Multiplying one side by cosine is not rearranging! You altered that side of the equation, and therefore you should have done the same to the other side.
This is a classic example of "assuming what you are trying to prove."
In algabraic proofs, especially when you are first being introduced to proofs, it is tempting to write down an equation then manipulate it until you get something that is "obviously" true. It cannot be stressed enough that this is not a proof, since what you assume (namely that the equality is true) may not be!
Here is a simple but enlightening example:
I claim I will prove -1=1.
Proof:
QED!
Now did I just rock the very foundation of mathematics by proving -1 = 1? Why not? If I have an equality I am allowed to square it as long as I do it to both sides, right?
The problem is that I started by assuming what I was trying to prove. I wanted to prove -1 = 1 and what I did was I wrote that down and did some math to both sides of the equation and ended up with an equality I know to be true. So I really didn't prove anything.
Start with one side of the equation (usually the more complicated side) and through the use of proven identities and simplification rearrange it into the other side. This constitutes a valid proof.
Furthermore that's exactly what happens when people bluntly follow various statements such as:When trying to prove an identity, think about what you're looking at: " an equality", which means that if you multiply one side of the equality by some expression, it will no longer be equal to the other side of the equality (that is of course if this expression does not evaluate to one). If you've decided to multiply one of the sides by some expression, you should also remember to divide this side by the same expression so as not to alter the equality you're trying to prove.When proving trig identities you don't let the sides interfere with each other.