What Are Pythagorean Identities?

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Most people remember the ​Pythagorean Theorem​ from beginner geometry — it's a classic. It's

a^2 + b^2 = c^2

where ​a​, ​b​ and ​c​ are the sides of a right triangle (​c​ is the hypotenuse). Well, this theorem can also be rewritten for trigonometry!

TL;DR (Too Long; Didn't Read)

TL;DR (Too Long; Didn't Read)

Pythagorean identities are equations that write the Pythagorean Theorem in terms of the trig functions.

The main ​Pythagorean identities​ are:

\sin^2(θ) + \cos^2(θ) = 1 \\ 1 + \tan^2(θ) = \sec^2(θ) \\ 1 + \cot^2(θ) = \csc^2(θ)

The Pythagorean identities are examples of ​trigonometric identities​: equalities (equations) that use trigonometric functions.

Why Does It Matter? 

The Pythagorean identities can be very useful for simplifying complicated trig statements and equations. Memorize them now, and you can save yourself a lot of time down the road!

Proof using the definitions of the trig functions

These identities are pretty simple to prove if you think about the definitions of the trig functions. For instance, let's prove that

\sin^2(θ) + \cos^2(θ) = 1

Remember that the definition of sine is opposite side / hypotenuse, and that cosine is adjacent side / hypotenuse.

So

\sin^2 = \frac{\text{opposite}^2} {\text{hypotenuse}^2}

And

\cos^2 = \frac{\text{adjacent}^2} {\text{hypotenuse}^2}

You can easily add these two together because the denominators are the same.

\sin^2 + \cos^2 = \frac{ \text{opposite}^2 + \text{adjacent}^2} {\text{hypotenuse}^2}

Now take another look at the Pythagorean Theorem. It says that ​a2 + ​b2 = ​c2. Keep in mind that ​a​ and ​b​ stand for the opposite and adjacent sides, and ​c​ stands for the hypotenuse.

You can rearrange the equation by dividing both sides by ​c2:

a^2 + b^2 = c^2 \\ \frac{a^2 + b^2}{ c^2 } = 1

Since ​a2 and ​b2 are the opposite and adjacent sides and ​c2 is the hypotenuse, you have an equivalent statement to the one above, with (opposite2 + adjacent2) / hypotenuse2 . And thanks to the work with ​a​, ​b​, ​c​ and the Pythagorean Theorem, you can now see this statement equals 1!

So

\frac{ \text{opposite}^2 + \text{adjacent}^2} {\text{hypotenuse}^2} = 1

and therefore:

\sin^2 + \cos^2 = 1

(And it's better to write it out properly: sin2(​θ​) + cos2(​θ​) = 1).

The Reciprocal Identities

Let's spend a few minutes looking at the ​reciprocal identities​ as well. Remember that the ​reciprocal​ is one divided by ("over") your number – also known as the inverse.

Since cosecant is the reciprocal of sine:

\csc(θ) = \frac{1}{\sin(θ)}

You can also think about cosecant using the definition of sine. For instance, sine = opposite side / hypotenuse. The inverse of that will be the fraction flipped upside-down, which is hypotenuse / opposite side.

Similarly, cosine's reciprocal is secant, so it's defined as

\sec(θ) = \frac{1}{\cos(θ)} \text{ or } \frac{\text{hypotenuse}}{\text{adjacent side}}

And tangent's reciprocal is cotangent, so

\cot(θ) = \frac{1}{\tan(θ)} = \frac{\text{adjacent side}}{\text{opposite side}}

The proofs for the Pythagorean identities using secant and cosecant are very similar to the one for sine and cosine. You can also derive the equations using the "parent" equation, sin2(​θ​) + cos2(​θ​) = 1. Divide both sides by cos2(​θ​) to get the identity 1 + tan2(​θ​) = sec2(​θ​). Divide both sides by sin2(​θ​) to get the identity 1 + cot2(​θ​) = csc2(​θ​).

Good luck and be sure to memorize the three Pythagorean identities!

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