I’m sure that this question has been asked in a million high school math offices, so apologies for the rudimentary nature of the question.
I’m teaching Precalculus for the first time. And I’m about to teach proving trig identities, like:
I understand that the standard ways to prove trig identities is:
(a) pick one side of the equation, and keep morphing it until it matches the second side of the equation
(b) individually modify both side of the equations independently until they equal the same thing.
I always learned that what you cannot do is start mixing both sides of the equations. So, for the equation above, you can’t cross multiply to get:
and keep on simplifying both sides to show they are the same and the equality is true.
The reasons I’ve heard this is not allowed:
1. Because I said so.
2. You can only cross multiply if you know the equality is true. But that’s precisely what you’re trying to prove. You are assuming the statement is true to prove the statement is true.
However, both explanations are unsatisfying to me. The first one is for obvious reasons. My objection with the second one is that it seems to always work for these problems. Although I know it is logically unsound, I can’t quite pinpoint why with a concrete example to demonstrate it..
My questions are the following:
What do you do to explain to your kids why you can only work the sides of the equality independently? Does it convince them?
Does anyone have a good example involving trigonometric identities that illustrates that bad things happen when you don’t solve the sides independently, but start mixing them together? Like proving something that isn’t true actually is true… or proving something true that actually isn’t true?
Thanks for any help. I feel a little foolish, like I’m missing something obvious. Like I should know this. But hey, if I knew everything, I wouldn’t need all y’all.