Lets say I have an equation like:
$$\frac{\sin^2x+\cos^2x}{\cos^2x\sec^2x}=1$$

Our teacher said you have to verify the equality by simplifying the left hand side or the right hand side without moving(doing operations to both sides) anything from one side of the equation to the other. 
I have a problem with this because I proposed that we can verify the equality by multiplying the denominator by 1 and moving it to the other side. Then you can proceed to verify the equation by substituting trig identities. My argument is that if correct algebraic manipulation is done, then the equality is the same. It won't suddenly become false or true. 

My method:
\*\*the addition to the classical method is that you can multiply the equation by both sides while preserving the equality which is specifically not multiplying by 0 or applying operators such as derivatives or squaring etc as these do not preserve equalities. 

$$\sin^2x+\cos^2x = \cos^2x·\sec^2x$$
$$1=\frac{\cos^2x·1}{\cos^2x}$$
$$1=1$$ equality is true

It requires ingenuity to be able to do the right manipulations...

Is my teacher justified in requiring that nothing is moved between the sides? is there a convincing argument for my side or am I wrong? 
I would be grateful if someone can enlighten me; just why is it wrong to do algebraic manipulation before verifying an equality? Keep in mind that this method is meant to be extended to the more complicated problems. I just gave a simple example to illustrate it.

For reiteration and clarity, my question is why is my method wrong? What evidence is available to back up any certain answer?