IB Maths AA 3.6 Notes

Core identities

A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variable for which both sides are defined. These identities are useful for simplifying expressions, proving results, and solving equations.

Some of the most important identities are:

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

$sin2x=2\sin x\cos x$

$\cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$

These are the main identities used throughout this topic.

This allows us to simplify trigonometric expressions in the same way as algebraic expressions.

• $3\cos x+4\cos x=7\cos x$
• $\tan\theta-3\tan\theta=-2\tan\theta$

This works because $\sin x$, $\cos x$ and $\tan x$ behave like algebraic quantities.

When proving an identity, work with one side only and transform it until it becomes the other side, just like a normal deductive proof. Do not start by assuming both sides are equal.

Show that $\cos x-\cos x\sin^2x=\cos^3x$.

Start with the left-hand side: $\cos x-\cos x\sin^2x=\cos x(1-\sin^2x)$.

Using $1-\sin^2x=\cos^2x$, this becomes $\cos x\cos^2x$.

So the expression simplifies to $\cos^3x$.

The identity is proven.

Show that $\frac{1-\cos2x}{\sin2x}=\tan x$.

Start with the left-hand side: $\frac{1-\cos2x}{\sin2x}$.

Using $\cos2x=1-2\sin^2x$, we get $1-\cos2x=2\sin^2x$.

Using $\sin2x=2\sin x\cos x$, the expression becomes $\frac{2\sin^2x}{2\sin x\cos x}$.

$\frac{\sin x}{\cos x}=\tan x$.

The identity is proven.