Alright, so I did this test at school, and I was asked to prove this identity:
So I went along my merry way:
and prepared to make it into
But being the clueless person I am, I included the intermediary step
which is wrong and loses marks even if it still reaches the answer.
The correct step in the given solution was to make a common denominator through:
And the thing is, I don't understand what the difference between the two methods is. As far as I can see, the given correct solution multiplies both sides of the fraction by
and I'm doing the same thing except multiplying by each cos(A) cos(B) one at a time instead of both at the same time. And as far as I can see, multiplying by two given numbers m and n is the same as multiplying by mn, right?
I asked my teacher about this and she said something about not accounting for cos(A) = 0. She justified the given correct solution with something like "the existence of tan(A) in the original solution automatically rules out cos(A) = 0 and that's why we can multiply both sides of it, but you made it into an addition and needed an extra condition 'for cos(A) = 0' in that line" (I don't remember exactly)
The proof concludes:
Could someone shed some light on this? Thanks.
The identity has certain constraints on the domains of both sides.
On the right, we see that A and B must differ, due to the denominator.
More specifically, A-B must not be any multiple of 180 degrees,
otherwise the right side is undefined.
On the left, we see again that A and B must differ.
However, tanA and tanB are both undefined for any odd multiple of 90 degrees.
If you place A=90 degrees and B=30 degrees for example into the right side,
you get a solution, but try placing those angle values into the left!
Hence, tanA and tanB are both undefined when cosA and cosB are zero.
Therefore, multiplying by cosA/cosA=cosB/cosB=(cosAcosB)/(cosAcosB)=1
is not an issue,
since cosA and cosB cannot be zero for the allowable domain of this identity.
If you are aware of this, then your method is fine.