Pythagorean Identities

Pythagorean Identities
Circle Identities

Trig identities relating sine with cosine, tangent with secant, and cotangent with cosecant. Derived from the Pythagorean theorem.

Pythagorean Identities

sin² θ + cos² θ = 1

tan² θ + 1 = sec² θ

cot² θ + 1 = csc² θ

Key Formula

\begin{gathered}\sin^2\theta + \cos^2\theta = 1\\\tan^2\theta + 1 = \sec^2\theta\\\cot^2\theta + 1 = \csc^2\theta\end{gathered}

Where:

$\theta$ = Any angle, measured in degrees or radians
$\sin\theta$ = Sine of the angle
$\cos\theta$ = Cosine of the angle
$\tan\theta$ = Tangent of the angle (sin θ / cos θ)
$\sec\theta$ = Secant of the angle (1 / cos θ)
$\cot\theta$ = Cotangent of the angle (cos θ / sin θ)
$\csc\theta$ = Cosecant of the angle (1 / sin θ)

Worked Example

Problem: Given that sin θ = 3/5 and θ is in the first quadrant, find cos θ, tan θ, and sec θ using Pythagorean identities.

Step 1: Start with the first Pythagorean identity and substitute sin θ = 3/5.

\sin^2\theta + \cos^2\theta = 1 \implies \left(\frac{3}{5}\right)^2 + \cos^2\theta = 1

Step 2: Solve for cos²θ.

\frac{9}{25} + \cos^2\theta = 1 \implies \cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}

Step 3: Since θ is in the first quadrant, cosine is positive. Take the positive square root.

\cos\theta = \frac{4}{5}

Step 4: Find tan θ using the ratio of sine to cosine.

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

Step 5: Verify using the second Pythagorean identity, and find sec θ.

\tan^2\theta + 1 = \sec^2\theta \implies \left(\frac{3}{4}\right)^2 + 1 = \frac{9}{16} + 1 = \frac{25}{16} \implies \sec\theta = \frac{5}{4}

Answer: cos θ = 4/5, tan θ = 3/4, and sec θ = 5/4.

Another Example

This example focuses on algebraic simplification using the third Pythagorean identity (cot²θ + 1 = csc²θ), rather than finding unknown trig values from a known one.

Problem: Simplify the expression: sin²θ · cot²θ + sin²θ.

Step 1: Factor out sin²θ from both terms.

\sin^2\theta \cdot \cot^2\theta + \sin^2\theta = \sin^2\theta\,(\cot^2\theta + 1)

Step 2: Apply the third Pythagorean identity: cot²θ + 1 = csc²θ.

= \sin^2\theta \cdot \csc^2\theta

Step 3: Since csc θ = 1/sin θ, replace csc²θ with 1/sin²θ.

= \sin^2\theta \cdot \frac{1}{\sin^2\theta} = 1

Answer: The expression simplifies to 1.

Frequently Asked Questions

How are the Pythagorean identities derived?

They come from the Pythagorean theorem applied to the unit circle. On the unit circle, any point is (cos θ, sin θ) with radius 1, so x² + y² = 1 gives sin²θ + cos²θ = 1. Dividing both sides of this identity by cos²θ produces tan²θ + 1 = sec²θ, and dividing by sin²θ produces cot²θ + 1 = csc²θ.

When do you use Pythagorean identities?

You use them whenever you need to convert between trig functions — for instance, rewriting an expression entirely in terms of sine and cosine, or eliminating a squared trig function during simplification. They appear constantly in calculus (especially integration of trig functions), physics, and proofs involving other trig identities.

What is the difference between the three Pythagorean identities?

All three are equivalent forms of the same underlying relationship. The first (sin²θ + cos²θ = 1) is the most fundamental. The second (tan²θ + 1 = sec²θ) pairs tangent with secant and is obtained by dividing the first identity by cos²θ. The third (cot²θ + 1 = csc²θ) pairs cotangent with cosecant and is obtained by dividing the first identity by sin²θ.

Pythagorean Identities vs. Reciprocal Identities

	Pythagorean Identities	Reciprocal Identities
What they relate	Squares of trig function pairs (e.g., sin²θ + cos²θ)	A trig function and its reciprocal (e.g., csc θ = 1/sin θ)
Number of identities	Three identities	Three identities (sin–csc, cos–sec, tan–cot)
Derived from	The Pythagorean theorem on the unit circle	Definitions of the six trig functions
Primary use	Eliminating or converting squared trig terms	Rewriting one trig function as another's reciprocal

Why It Matters

Pythagorean identities appear in nearly every trigonometry, precalculus, and calculus course. You need them to simplify integrals (like converting ∫cos²θ dθ), solve trig equations, and verify other identities. In physics and engineering, they are essential for resolving vector components and analyzing wave equations.

Common Mistakes

Mistake: Writing sin²θ − cos²θ = 1 instead of sin²θ + cos²θ = 1.

Correction: The identity always uses addition: sin²θ + cos²θ = 1. The expression sin²θ − cos²θ equals −cos 2θ, which is a double-angle identity, not a Pythagorean identity.

Mistake: Forgetting to consider the sign (positive or negative) when taking a square root.

Correction: When you solve cos²θ = 16/25, the result is cos θ = ±4/5. You must use the quadrant of θ to determine the correct sign. For example, cosine is negative in quadrants II and III.

Related Terms

Trig Identities — Broader category containing the Pythagorean identities
Pythagorean Theorem — The geometric theorem from which these identities are derived
Sine — Appears in the first Pythagorean identity
Cosine — Paired with sine in the first identity
Tangent — Paired with secant in the second identity
Secant — Paired with tangent in the second identity
Cotangent — Paired with cosecant in the third identity
Cosecant — Paired with cotangent in the third identity