When solving trigonometric identities, you aren't allowed to work on both sides of the equation at once. The reason for this is that if you do arrive at a valid conclusion, it doesn't provide the validity of the initial equation - it just proves that if the initial equation is true, you can arrive at a valid equation.
I have a number of questions about this:
1) Why can't adding or subtracting to both sides by allowed? Regardless of the relation between the sides ($=$,$<$,$>$,$≤$ or $≥$), adding and subtracting doesn't change the relation. Unlike dividing or multiplying by a negative number (which inverses the sign), adding or subtracting doesn't change the sign.
2) If you do work on both sides, and you do arrive at a valid equation (i.e. the Pythagorean identity), can't you prove the initial equation by working backwards? Through this reasoning if you reach a valid equation by working on both sides, the initial equation is valid.
For example: Let $x$ represent a trigonometric identity you are checking the validity of. Let $y$ represent a proven identity such as the Pythagorean identity.
Let's you say you do the following:
i) Divide both sides of $x$ by $a$
ii) Multiply both sides of $x$ by $b$
iii) Add $c$ to both sides of $x$
iiii) You arrive at equation $y$
My math teacher would argue this doesn't prove the validity of $x$ - it simply proves that if $x$ is true, you can arrive at $y$. However, if you start at $y$, and do the above steps backward (Subtract $c$, divide by $b$ and multiply by $a$) won't you arrive at $x$? Therefore if you do arrive at $y$ through working on both sides of $x$, shouldn't $x$ be valid since you can arrive at $x$ by working backwards starting at $y$?
3) Some trigonometric identities are extremely complicated and take a while to solve. How do you tell the difference between you not being able to find the proof and when the equation is not true? Because it would be a waste of time trying to find the proof of an trigonometric identity that is invalid.