Students arrive at trig exams clutching an identity sheet they cannot use, because the strategy of "scan for a matching pattern" fails when the problem requires three or four transformations before any identity applies. The productive approach is to understand where each identity comes from, which reduces the sheet from a list of thirty disconnected formulas to six categories with underlying logic.
Category 1 — Pythagorean identities
These three identities derive from the unit circle definition of sin and cos, combined with the Pythagorean theorem.
sin²θ + cos²θ = 1 (the fundamental one — every other trig identity can be derived from this) 1 + tan²θ = sec²θ (divide both sides of the first by cos²θ) cot²θ + 1 = csc²θ (divide both sides of the first by sin²θ)
You only need to memorise the first one. The other two come from dividing through by sin² or cos².
Category 2 — Double angle formulas
These come from the angle addition formulas with the two angles set equal to each other.
sin(2θ) = 2 sinθ cosθ cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
The three versions of the cos double angle formula come from substituting the Pythagorean identity into the first version. The first version is fundamental; the others are derived.
Category 3 — Half-angle formulas
These come from the double angle formulas, solved for sin²θ and cos²θ.
sin²θ = (1 − cos(2θ)) / 2 cos²θ = (1 + cos(2θ)) / 2
Useful in integration when you encounter sin² or cos² — replace them with these half-angle forms, which integrate easily.
Category 4 — Product-to-sum formulas
These come from adding and subtracting the angle addition formulas. They appear in more advanced integration and Fourier analysis.
sinA cosB = ½[sin(A+B) + sin(A−B)] cosA cosB = ½[cos(A+B) + cos(A−B)] sinA sinB = ½[cos(A−B) − cos(A+B)]
On an exam, if you see a product of two trig functions that does not simplify via double angle, reach for this category.
Category 5 — Sum-to-product formulas
The reverse of category 4. Useful when you need to factor a sum of trig functions.
sinA + sinB = 2 sin((A+B)/2) cos((A−B)/2)
Category 6 — Reciprocal and quotient identities
These are definitions, not derived identities:
tan = sin/cos, cot = cos/sin, sec = 1/cos, csc = 1/sin.
Students sometimes treat these as surprising results rather than definitions. They are not surprising — they are how these functions are defined in terms of sin and cos.
The exam strategy
When stuck: convert everything to sin and cos using the category 6 identities. This always gives you something to work with, even if it is algebraically messier. Then look for Pythagorean identity applications (category 1) to simplify.
If the problem involves double angles or half-angles, check whether the argument of a trig function is twice or half the argument you want (categories 2 and 3).
If the problem involves a product where you expect a sum (common in integration), use category 4.
Frequently Asked Questions
Do I need to memorise all trig identities for AP Calculus?
AP Calculus provides a formula sheet that includes most identities you need. Understanding where they come from is more valuable than memorising them, because you can derive any forgotten identity from the fundamental ones during the exam.
Which trig identity is used most in calculus integration?
The half-angle formulas (sin²θ = (1 − cos 2θ)/2 and cos²θ = (1 + cos 2θ)/2) are used constantly in integrating powers of sin and cos. The Pythagorean identity (sin²θ + cos²θ = 1) is the most generally used single identity.