Trigonometric Identities

Every time your phone tidies up a noisy call, a music app strips the hiss from a track, or an engineer tames the current humming through the power grid, trig identities are quietly at work — folding tangled combinations of waves into something simple enough to handle. They all start from one short, surprising fact.

Here is a small miracle. Pick any angle you like — 17^\circ, 90^\circ, 1000^\circ, a random number your calculator spits out. Square its sine, square its cosine, add them. You will always get exactly 1. Not roughly, not usually — always, forever, for every angle that has ever existed.

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

A statement like this — true for every value of the variable — is called an identity. It's a different beast from an ordinary equation. When you meet 2x = 6 you solve it for the one value that works (x = 3). An identity needs no solving: it is a permanent, always-true fact, and that makes it a tool. Wherever a messy \sin^2\theta + \cos^2\theta appears, you may simply cross it out and write 1. The two workhorses of the whole subject are the one above and its partner,

\tan\theta = \frac{\sin\theta}{\cos\theta}.

Get comfortable with these two and you can bend almost any trig expression into a friendlier shape.

Sine and cosine live on the unit circle

Where does \sin^2\theta + \cos^2\theta = 1 come from? Draw a circle of radius 1 — the unit circle — centred at the origin. A point P on the circle at angle \theta (measured from the positive x-axis) has coordinates

P = (\cos\theta,\ \sin\theta).

That point sits exactly distance 1 from the centre. Drop a vertical line from P to the x-axis and you get a right-angled triangle with horizontal side \cos\theta, vertical side \sin\theta, and hypotenuse 1. Now Pythagoras on that triangle gives

(\cos\theta)^2 + (\sin\theta)^2 = 1^2,

which is our identity, written the way mathematicians prefer: \sin^2\theta + \cos^2\theta = 1. It really is just Pythagoras' theorem wearing a disguise.

See it on the unit circle

Step through the figure: a radius to P, then the right triangle whose sides are \cos\theta and \sin\theta with hypotenuse 1. The Pythagorean identity is the last line to appear — because it is the triangle.

Tangent is just their ratio

Tangent isn't a third mystery to memorise — it's sine divided by cosine:

\tan\theta = \frac{\sin\theta}{\cos\theta}.

And because the Pythagorean identity ties \sin\theta and \cos\theta together, all three ratios belong to one family: give me any one of them (and a quadrant) and I can hand you the other two. That single-family idea is what makes the identities so powerful — they are rearrangements of the same fact. The most useful rearrangements to keep in your pocket:

\sin^2\theta = 1 - \cos^2\theta, \qquad \cos^2\theta = 1 - \sin^2\theta.

\sin^2\theta + \cos^2\theta = 1 — the Pythagorean identity;
\tan\theta = \dfrac{\sin\theta}{\cos\theta};
rearranged: \sin^2\theta = 1 - \cos^2\theta and \cos^2\theta = 1 - \sin^2\theta;
so given any one ratio you can find the others.

Worked example 1 — simplify an expression

Simplify \dfrac{\sin^2\theta}{1 - \cos^2\theta}.

Look at the denominator. By the rearranged identity, 1 - \cos^2\theta = \sin^2\theta. Swap it in:

\frac{\sin^2\theta}{1 - \cos^2\theta} = \frac{\sin^2\theta}{\sin^2\theta} = 1.

A frightening-looking fraction collapses to the number 1 — no calculator, no angle, just the identity. That collapse is exactly what engineers exploit to tame the trig inside wave and circuit equations.

Worked example 2 — prove an identity

Prove that \tan\theta\,\cos\theta = \sin\theta. The golden rule of proving an identity: start with one side and manipulate it until it turns into the other. Never shove terms across the \equiv sign as if you were solving — you'd be assuming the very thing you're trying to prove.

Take the left-hand side and replace \tan\theta with its definition:

\tan\theta\,\cos\theta = \frac{\sin\theta}{\cos\theta}\cdot\cos\theta = \sin\theta.

The \cos\theta cancels, we land on the right-hand side, and the proof is done. One side in, the other side out.

Worked example 3 — find a missing ratio

Given \sin\theta = \tfrac{3}{5} with \theta acute, find \cos\theta and \tan\theta.

Use the Pythagorean identity rearranged for \cos^2\theta:

\cos^2\theta = 1 - \sin^2\theta = 1 - \left(\tfrac{3}{5}\right)^2 = 1 - \tfrac{9}{25} = \tfrac{16}{25}.

So \cos\theta = \tfrac{4}{5} (positive, because \theta is acute — take the + root). Then tangent falls straight out of the ratio:

\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \frac{3}{4}.

The classic 3-4-5 triangle, recovered from a single ratio using nothing but the identities.

Three traps that quietly cost real marks:

\sin^2\theta means (\sin\theta)^2 — the sine, then squared. It is not \sin(\theta^2) (the sine of the angle squared). The little index sits up by the "sin" purely to avoid the ugly brackets of (\sin\theta)^2. Read it as "sine-squared", and never feed \theta^2 into your calculator.
An identity is not an equation. An identity (write it with \equiv) is true for all \theta and needs no solving. An equation like \sin\theta = 0.5 is true only for special values, which you go and find — that's the whole job of solving trig equations.
When proving, work one side only. Massage the left-hand side (or the right) until it becomes the other. Don't "add \cos^2\theta to both sides" or move terms across — that treats an unproven claim as if it were already true.

Because \sin^2\theta + \cos^2\theta = 1 is a seed. Divide every term by \cos^2\theta and out pops \tan^2\theta + 1 = \sec^2\theta; divide instead by \sin^2\theta and you get 1 + \cot^2\theta = \csc^2\theta. Combine it with the angle-addition formulas and you can unfold double-angle identities, half-angle identities, and the rules that turn a sum of waves into a product.