Trigonometric identities: the three you memorise and the rest you derive

Optional Mathematics students try to memorise twenty identities and forget all of them in the exam. There are three worth memorising. Everything else comes from those three in under a minute, and deriving it is safer than remembering it.

The three:

1. sin squared A + cos squared A = 1. This is Pythagoras on the unit circle, nothing more.

2. sin(A + B) = sin A cos B + cos A sin B.

3. cos(A + B) = cos A cos B minus sin A sin B.

Now watch what falls out.

Divide identity 1 by cos squared A: tan squared A + 1 = sec squared A. Divide it by sin squared A instead: 1 + cot squared A = cosec squared A. Two identities from one, and you cannot get the signs wrong because you derived them.

Put B = A in identity 2: sin 2A = 2 sin A cos A. Put B = A in identity 3: cos 2A = cos squared A minus sin squared A, and using identity 1 that becomes 2 cos squared A minus 1, or 1 minus 2 sin squared A. That is the double angle set, all of it.

Replace B with minus B in identities 2 and 3, remembering sin(minus B) = minus sin B and cos(minus B) = cos B, and you have the subtraction formulas.

Divide identity 2 by identity 3 and divide top and bottom by cos A cos B: tan(A + B) = (tan A + tan B) over (1 minus tan A tan B).

The half angle formulas are the double angle formulas read backwards with A replaced by A/2.

So the method in the exam: write the three at the top of your rough page the moment you sit down. When a question needs an identity you do not remember, derive it there. It takes a minute and it cannot be wrong.

The common error is not the identities, it is algebra afterwards: cancelling a term that appears on both sides of a product instead of a sum, or dividing by something that could be zero. Go slowly on the algebra; the trigonometry is the easy part.

Worked examples for each of these are in the Class 9 and Class 10 Optional Maths notes I uploaded.