Trigonometric identities

What an identity is

• An identity is a statement that is true for every value of the variable — not an equation to be solved.
• $\sin^2\theta + \cos^2\theta = 1$ holds whatever $\theta$ is. By contrast, $\sin\theta = 0.5$ is an equation, true only for particular angles.
• Identities are tools. Their purpose is to let you rewrite an expression into a form you can work with.

The Pythagorean identities

$$\sin^2\theta + \cos^2\theta = 1$$

• This is the foundational one, and it comes from Pythagoras applied to a point on the unit circle: the coordinates are $(\cos\theta, \sin\theta)$ and the radius is 1.

• The two rearrangements matter as much as the original:

$$\sin^2\theta = 1 - \cos^2\theta \qquad \cos^2\theta = 1 - \sin^2\theta$$

• Dividing through by $\cos^2\theta$ gives a second Pythagorean identity:

$$1 + \tan^2\theta = \sec^2\theta$$

• Dividing through by $\sin^2\theta$ gives a third:

$$1 + \cot^2\theta = \csc^2\theta$$

• Note the pattern: each has a $1$, a squared function, and the square of its reciprocal partner.

The quotient identities

$$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}$$

• These are how you convert a tangent into sines and cosines, which is often the step that makes an expression simplify.
• A useful strategy: when an expression mixes several functions, rewrite everything in terms of $\sin$ and $\cos$ and see what cancels.

The compound angle identities

$$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$$

$$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$$

$$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$$

• Watch the signs on the cosine identity. They are opposite to the sign on the left: $\cos(A + B)$ has a minus in the middle. This is the most-missed detail on the page.
• The $\mp$ symbol means "the opposite sign to the one above".

The double angle identities

• Setting $B = A$ in the compound identities gives:

$$\sin 2A = 2\sin A\cos A$$

$$\cos 2A = \cos^2 A - \sin^2 A$$

• The cosine version has three equivalent forms, obtained by applying the Pythagorean identity:

$$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$$

• Choosing the right form is the skill. If an expression contains only cosines, use $2\cos^2 A - 1$; if only sines, use $1 - 2\sin^2 A$.
• Rearranged, these give the forms needed for integration at Level 3:

$$\cos^2 A = \frac{1 + \cos 2A}{2} \qquad \sin^2 A = \frac{1 - \cos 2A}{2}$$

Proving an identity

• The rule: start with one side and transform it until it becomes the other. Do not move terms across the equals sign or operate on both sides, because that assumes the very thing you are proving.
• Start with the more complicated side — there is more to work with.
• Common tactics:
  • Convert everything to $\sin$ and $\cos$
  • Find a common denominator and combine fractions
  • Look for a Pythagorean substitution, especially $1 - \cos^2$ or $1 - \sin^2$
  • Factorise, particularly a difference of two squares
• State clearly which side you are working on, and finish by saying the two sides are now equal.

Simplifying expressions

• The same tactics simplify expressions that are not identities:
  • $$\dfrac{\sin\theta}{\tan\theta} = \dfrac{\sin\theta}{\sin\theta/\cos\theta} = \cos\theta$$
  • $$\dfrac{1 - \cos^2\theta}{\sin\theta} = \dfrac{\sin^2\theta}{\sin\theta} = \sin\theta$$