Trigonometric Identities: Why sin²θ + cos²θ = 1 Had to Be True

Trigonometric Identities: Why sin²θ + cos²θ = 1 Had to Be True | Ideasthesia

Trig identities are not arbitrary facts. They're geometric necessities.

Here's the unlock: sin²θ + cos²θ = 1 is the Pythagorean theorem. It's saying that the point (cos θ, sin θ) sits on the unit circle, where x² + y² = 1. The identity isn't a trig fact — it's a circle fact wearing trig notation.

Every trig identity works this way. They look like algebra, but they're really geometry. Once you see where they come from, you stop memorizing and start deriving.

The Pythagorean Identity

sin²θ + cos²θ = 1

Why? On the unit circle, any point has coordinates (cos θ, sin θ). Every point on the unit circle satisfies x² + y² = 1.

Therefore: cos²θ + sin²θ = 1. ∎

This is the most important trig identity. All others flow from it or from the definitions.

The Other Pythagorean Identities

Divide sin²θ + cos²θ = 1 by cos²θ:

tan²θ + 1 = sec²θ

(Using tan θ = sin θ / cos θ and sec θ = 1/cos θ)

Divide by sin²θ:

1 + cot²θ = csc²θ

Three Pythagorean identities, all from one circle equation.

Reciprocal Identities

These follow directly from definitions:

csc θ = 1/sin θ (cosecant is the reciprocal of sine) sec θ = 1/cos θ (secant is the reciprocal of cosine) cot θ = 1/tan θ = cos θ/sin θ (cotangent is the reciprocal of tangent)

Nothing to derive here — these are just names for reciprocals.

Quotient Identities

tan θ = sin θ / cos θ cot θ = cos θ / sin θ

These come from the geometry: tan is y/x on the unit circle, sin is y, cos is x.

Cofunction Identities

sin(90° - θ) = cos θ cos(90° - θ) = sin θ tan(90° - θ) = cot θ

Why? Complementary angles (summing to 90°) swap the roles of opposite and adjacent in a right triangle.

If θ and (90° - θ) are the two acute angles of a right triangle:

The side opposite θ is adjacent to (90° - θ)
The side adjacent to θ is opposite (90° - θ)

So sin θ (opposite/hypotenuse) equals cos(90° - θ) (adjacent/hypotenuse for the complement).

This is why "cosine" is "co-sine" — the sine of the complementary angle.

Even-Odd Identities

cos(-θ) = cos θ (cosine is even) sin(-θ) = -sin θ (sine is odd) tan(-θ) = -tan θ (tangent is odd)

Why? Reflecting across the x-axis takes (cos θ, sin θ) to (cos θ, -sin θ). The x-coordinate stays the same. The y-coordinate negates.

Sum and Difference Formulas

sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B

cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B

These are harder to derive but geometrically real. They describe what happens when you rotate by angle A and then by angle B.

Rotation by A followed by rotation by B equals rotation by (A + B). The formulas express this in coordinates.

Double Angle Formulas

Set A = B in the sum formulas:

sin(2θ) = 2 sin θ cos θ

cos(2θ) = cos²θ - sin²θ

Using sin²θ + cos²θ = 1, you can write cos(2θ) three ways:

cos²θ - sin²θ
2cos²θ - 1
1 - 2sin²θ

Half Angle Formulas

Solve the double angle formulas for the single angles:

sin(θ/2) = ±√[(1 - cos θ)/2] cos(θ/2) = ±√[(1 + cos θ)/2]

The ± depends on which quadrant θ/2 is in.

Product-to-Sum Formulas

These convert products of trig functions into sums:

sin A cos B = ½[sin(A+B) + sin(A-B)] cos A cos B = ½[cos(A+B) + cos(A-B)] sin A sin B = ½[cos(A-B) - cos(A+B)]

Useful for integration and signal processing.

Sum-to-Product Formulas

The reverse direction:

sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2) cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)

These help simplify sums of trig functions.

Why These Identities Exist

Every identity reflects some geometric fact:

Pythagorean identities: Points on the circle satisfy x² + y² = 1
Even/odd identities: Reflection symmetry of the circle
Cofunction identities: Complementary angles swap x and y
Sum formulas: Rotation composition
Double angle: Rotation by 2θ = rotation by θ twice

The identities aren't arbitrary. They're the algebra of circle geometry.

Using Identities for Simplification

Example: Simplify sin²θ/(1 - cos θ)

Use sin²θ = 1 - cos²θ = (1 - cos θ)(1 + cos θ):

sin²θ/(1 - cos θ) = (1 - cos θ)(1 + cos θ)/(1 - cos θ) = 1 + cos θ

Proving Identities

To prove a trig identity, transform one side into the other using known identities.

Strategy:

Work with the more complicated side
Convert everything to sine and cosine
Look for Pythagorean substitutions
Factor and simplify

Example: Prove tan θ + cot θ = sec θ csc θ

Left side: sin θ/cos θ + cos θ/sin θ = (sin²θ + cos²θ)/(sin θ cos θ) = 1/(sin θ cos θ) = (1/cos θ)(1/sin θ) = sec θ csc θ ✓

The Core Insight

Trig identities are circle geometry in algebraic form.

sin²θ + cos²θ = 1 says points lie on a unit circle. The sum formulas describe rotation composition. The even/odd identities capture reflection symmetry.

When you see a trig identity, ask: what geometric fact does this express? Once you see the geometry, the algebra becomes memorable — and derivable.

Don't memorize identities. Understand circles.

Part 9 of the Trigonometry series.

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