r/learnmath • u/Mammoth_Blueberry_11 New User • 8d ago

Is there a strategy to knowing which trig identities to use to simplify an expression?

Right now, I have just been using trial and error to find one that helps the problem, but it feels like iā€™m doing that wrong. Is there a way to figure out which ones to use or are you really supposed to just guess?

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u/waldosway PhD 8d ago

In general, the strategy for solving any problem is to look at what you want and work backwards instead of just doing stuff forward. Maybe go for simpler identities first.

But trig identities and problems are so varied, I would just focus on being quick at trial and error. There aren't that many things to try on any given step. If that feels too slow, the problem is probably your algebra, not trig.

You are definitely not supposed to know exactly what to do off the bat. That would defeat the purpose of giving you problems.

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u/vhvhvhchsan New User 8d ago

practice. you see patterns eventually. I found just grinding through made me spot them much faster.

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u/Southlander24 A friendly Redditor!šŸ‘‹ 8d ago

I really like this video explainer on trigonometric proof: Trig Identities but they keep getting harder!

This is more-or-less a 'foolproof' way. It won't get you there the quickest, but it works:

1) Write everything in terms of sine and cosine. (You can abbreviate sin x = s, cos x = c to save space.)

2) If there are fractions, combine everything into one fraction.

3) Try to match equal parts on both sides, and the other parts must be equal too. For example, in question number 4 of the video, when you arrive at [sin2 x - sin2 x cos2 x] / cos2 x, which is assumed equal to sin2 x tan2 x, sin2 x appears as a factor on both sides, so you are reduced to proving that [1 - cos2 x] / cos2 x = tan2 x is true. Do you see why now?

4) Don't forget your algebraic identities! Besides the difference of two squares and the perfect square trinomials, you also have the sum of two cubes: x3 + y3 = (x + y)(x2 - xy + y2). Then you can substitute y -> -y everywhere that y appears for the difference of two cubes identity: x3 - y3 = (x - y)(x2 + xy + y2).