MISSION · Geometry
Trigonometric Identities
Equations that are true for every angle, and what they're for
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LEVEL 2 · LEARN
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The idea in one sentence
An identity is true for every angle, not just some — so it can be used to rewrite any expression, anywhere, without checking conditions.
The Pythagorean identity
\[\sin^2\theta + \cos^2\theta = 1\]This is the circle’s equation in disguise. On the unit circle a point is \((\cos\theta, \sin\theta)\), and its distance from the centre is 1, so \(x^2 + y^2 = 1\) becomes exactly the identity above.
Note the notation: \(\sin^2\theta\) means \((\sin\theta)^2\), not \(\sin(\theta^2)\). The power is applied to the answer.
Rearranged, it gives two more forms you’ll reach for constantly:
\[\sin^2\theta = 1 - \cos^2\theta \qquad \cos^2\theta = 1 - \sin^2\theta\]The quotient identity
\[\tan\theta = \frac{\sin\theta}{\cos\theta}\]Straight from the triangle definitions: opposite over hypotenuse, divided by adjacent over hypotenuse, leaves opposite over adjacent. The hypotenuse cancels.
It also explains why \(\tan 90°\) is undefined — \(\cos 90° = 0\), so the fraction divides by zero.
The addition and double-angle formulas
\(\sin(A + B) = \sin A \cos B + \cos A \sin B\) \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
Note the minus sign in the cosine formula. It is not a misprint, and it is the detail most often lost.
Setting \(B = A\) gives the double-angle formulas:
\[\sin 2A = 2\sin A\cos A \qquad \cos 2A = \cos^2 A - \sin^2 A\]And the most important consequence of all: \(\sin 2A\) is not \(2\sin A\). Trigonometric functions do not distribute over multiplication or addition. \(\sin(A + B) \ne \sin A + \sin B\), and testing it with \(A = B = 30°\) settles the matter — \(\sin 60° = 0.866\) while \(\sin 30° + \sin 30° = 1\).
What identities are for
Simplifying. \(\frac{\sin^2\theta + \cos^2\theta}{\cos\theta}\) collapses to \(\frac{1}{\cos\theta}\) immediately.
Solving equations. An equation with both \(\sin^2\) and \(\cos^2\) in it can be rewritten in terms of one function, at which point it becomes an ordinary quadratic.
Integration. Many integrals are impossible until an identity turns the integrand into something standard — \(\cos^2 x\) is integrated by first rewriting it using the double-angle formula.
Proving an identity
The method is fixed: work on one side only until it matches the other. Usually start with the messier side, convert everything into sines and cosines, and simplify.
What you must not do is move terms across the equals sign as though solving an equation. That assumes the identity is true, which is what you were asked to demonstrate.
Common mix-ups
- Distributing across the function. \(\sin 2A \ne 2\sin A\).
- Reading \(\sin^2\theta\) as \(\sin(\theta^2)\).
- Losing the minus in the cosine addition formula.
- Rearranging across the equals sign while proving.
Check yourself
Simplify \(\frac{\sin 2\theta}{\sin\theta}\), and explain why \(\sin^2\theta = 1 - \cos^2\theta\).
\(\frac{2\sin\theta\cos\theta}{\sin\theta} = 2\cos\theta\). The second is the Pythagorean identity rearranged, which itself is the unit circle’s \(x^2 + y^2 = 1\).
LEVEL 3 · REFLECT
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