Trig Identity Proofs: Four Ways In When You Are Stuck

Proving an identity is not solving an equation — you cannot operate on both sides at once, because that assumes the thing you are trying to prove. Pick one side and transform it into the other. Which side? Almost always the one that looks more complicated, because simplifying is easier than building complexity. Write the target side in the margin and keep looking at it — it tells you which direction to push.

When the expression mixes tan, cot, sec and csc, the most reliable first move is to rewrite them all in sin and cos: tan x = sin x / cos x, sec x = 1 / cos x, and so on. Once converted, a structure you can combine over a common denominator, cancel, or hit with sin²x + cos²x = 1 usually appears. It is not always the fastest route, but when you have no idea where to start, it almost always opens the problem up.

Every step of a proof leans on these, and there is no time to re-derive them under exam pressure: • Pythagorean: sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = csc²x • Double angle: sin 2x = 2 sin x cos x, cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1 • Compound angle: the expansions of sin(A ± B) and cos(A ± B) The three forms of cos 2x matter most — a proof often needs you to pick whichever version lands closest to the target.

If pushing from one side to the other simply will not go, switch strategy: simplify the left-hand side and the right-hand side separately, each to some intermediate form. If the two intermediate forms are identical, the proof stands — just state "LHS = RHS, hence proved" at the end. This helps most when both sides are complicated and it is not obvious which turns into which. The cost is more steps and more room for mess, so work it in two columns on your rough paper and copy it up cleanly afterwards.

Now go practise

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