If your trig course was like mine, at some point you’ll be asked to prove one side of an equation equals another side, simplify a trig forumula, etc., using identities.

Of course, this is true outside of trig too…so my recommended approaches work beyond just this one subset of mathematics.

Now, I used the word “maze” deliberately. If you go into a real world maze (or do a maze puzzle), sometimes you’ll go the wrong way…have to back up…and try again. That’s true with these problems too. Don’t feel defeated if your first, second, etc. attempt fails. Just back up and try again!

And, understand that it’ll become easier and easier over time, where just looking at a problem will give you an idea of what the best steps likely are.

What should you try (numbering does *not* imply order or importance):

- For a proof: Choose one side and work it to the other side (you can accomplish a proof working both sides, but there is a good chance your teacher will require choosing a single side, and you are more likely to get lost in the maze if you don’t)
- For a proof: Choose the most complex side (think of ${\sin ^2}x + {\cos ^2}x = 1$…would you really want to make the right side become the left?)
- Don’t forget you can use all the algebra tricks you’ve learned before; the scary trig functions don’t change how they work (e.g. if you can handle factoring ${x^2} – {y^2} = (x + y)(x – y)$, you can handle factoring $1 – 4{\sin ^2x})$
- If you are dealing with fractions, find the common denominator
- Multiply by one (which either means multiplying what you have by a value over itself or both sides of an equation by the same value)
- Multiply by the conjugate (e.g. if you have $1 + \sin x$, multiply by $\frac{1-\sin x}{1-\sin x}$); this is a form of “multiply by one”
- Convert the equation into $\sin$ and $\cos$; e.g. if you are given $\tan$ and $\csc$, see what happens if you use $\frac{{\sin }}{{\cos }}$ and $\frac{1}{{\sin }}$ instead
- Watch for patterns; e.g. if you see ${\sin ^2}x$ or ${\cos ^2}x$, you should ask yourself, “Will the Pythagorean Identity help?”…or if you see something squared minus something squared, will factoring it to the equivalent of $(x + y)(x – y)$ help?
- Use identities, they are tools for converting something you don’t know how to handle into something you do
- Use other methods of factoring, especially if there is a denominator that might cancel out, at least in part
- If you have something that can be multiplied out, try it and and see what might disappear when you put all the components together, sometimes problems are written to cause pieces to cancel out or the result to be more manageable

Note that there is quite a bit of overlap between the approaches. #6 is a form of #5. #7 will often use #9. Some of the patterns you should look for in #8 are identities from #9. And so on…

So, what do you think? Any other advice you would give folks based on your experiences?

P.S. The picture is from an Indiana University Supplemental Instruction session where I was discussing this…and, as you can see, the list is growing… 🙂

P.P.S. #10 and #11 added on 18 February 2023

## Leave a Reply