Proving Trig Identities by Trying Stuff

If you remember your high-school geometry, you might recall being flummoxed by the "fill in the blanks" proofs you were supposed to do. They gave you the "givens" at the top, whatever it was that you were supposed to prove at the bottom, and a few blank lines in between. What the heck was supposed to go in those lines?

A good technique, I've found, is to cover up the blank lines, leaving just the givens and the thing to prove. Then think, "Okay, what do these givens make me think of? What do they feel like?" And put something down on the paper (that was covering over the blank lines). Then see where that would take you. Soon enough, you'd have proven what you were supposed to prove. And, almost always, when you pulled that paper aside, you'd note that you just so happened to take as many steps as they'd expected you to, and now you could fill in the blanks.

Why did that work? In large part, it was by design. The authors knew what they'd taught you already, and they'd given you a proof that was just a very small step forward from that. There may have been more than one way to prove it, but they were consciously steering you, so the steps you came up with were the steps that they'd intended you to take. And, once students see this, they are able to get over that psychological block.

The point I'd like to make here is that those proofs were completed by a process of, well, trying stuff. If something doesn't seem to be working, then backing up and trying something else. Otherwise, keep going. You can do this; you do know how. You just need to be willing to try stuff, and have faith in yourself.

• Prove the identity sin⁴(x) − cos⁴(x) = 2sin²(x) − 1

(By the way, there is no mathematical significance to the square brackets; I just wanted to use grouping symbols that wouldn't get lost in all the parentheses.)

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• Prove the following identity:

  (1 − cos²(α))(1 + cos²(α)) = 2sin²(α) − sin⁴(α)

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This, so you know, is how the textbook authors come up with those magical proofs, where you wonder how on earth they ever came up with one or another of the steps: they worked both sides until they made them match, and then rewrote their steps, working down one side to the matching step, and then back up with other side.

There will be times you'll need to do this. But, for your hand-in work, make sure you do the work the right way: Work down one side to the meeting place, and then hop over to the other side and work back up to the starting place. Only by showing your steps in that way will your proof be valid. But, hey — now you can do magic, too! 😜

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• Prove the following identity:

  sin²(θ)sec²(θ) + sin²(θ)csc²(θ) = sec²(θ)

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In addition to the "working on the more complicated side" and the "converting to sines and cosines" tricks, there is one other trick that hardly ever comes up, but it would probably be wise to familiarize yourself with it....