My question is about the rules of proving trig identities rather than showing that both sides are equal to each other.
For example for a problem like this:
$(sin(x) + cos(x))^2 = 1 + sin (2x)$
From what I've been told, you start with one side and by only using that one side you must obtain the other side. And this is how these problems are done.
So do you have to do this for both sides for it to be a proof?
Are you allowed to move terms from the RHS to the LHS and vice versa?
Again, I'm not asking how to do the problem as I already know this, but I want to know the rules involved when proving a trig identity problem.
Thanks.
Best Answer
If you work only with the LHS and manage to simplify it to the point where it's identically equal to the RHS, then you're done.
Similarly, If you work only with the RHS and manage to simplify it to the point where it's identically equal to the LHS, then you're done.
But there are other options.
For example, you could simplify the LHS to reach an expression $A$, and then, if you manage to simplify the RHS to also reach $A$, then you're done, since you've proved both LHS and RHS are equal to $A$.
What you're not allowed to do is start with the claimed identity, and reduce the equation to some equation which is clearly true, unless . . .
Unless what?
Unless each step of the reduction is reversible. As long as the steps are reversible, you can start with the goal and try to reduce it to a known identity. If you choose this approach, to make the reversibility clear to the reader, each step should be connected to the previous one by the symbol $\iff$ (if and only if).