I've just got a question regarding notation that I am not sure how to answer. I've been tasked with showing that the left hand side and right hand side of an equation equal each other, which is a enough simple task. However, I am not sure what notation I should be using so that what I am writing is still true. The problem is as follows: Show that $$ \frac{d^2T(t)}{dt^2}+\omega^2T(t)=0 $$ is satisfied by the given solution $$ T(t)=A\cos{\omega t}+B\sin{\omega t} $$ This is obviously not very difficult, but in class lectures my professors simply took the derivatives, substituted them into the left hand side and equated them to the right hand side immediately, before showing that it equals zero, and simply place a question mark above the equal sign. For example, the substitution would look something like: $$ (-A\omega^2 \cos{\omega t}-B\omega^2\sin{\omega t})+\omega^2(B\sin{\omega t}+A\cos{\omega t})\stackrel{?}{=} 0 $$
Is this proper notation, or is there a better way to notate this? It is just a minor thing that has been bugging me. For the record, I am an engineering student so my engineering professors are often not very careful with their notation. Thank you in advance for the help!
I think it's quite acceptable to argue in the form: We have $$\text{LHS} =\dots = stuff,$$ while $$\text{RHS} = \dots = stuff,$$ hence equality. $\square$