# What is the pythagorean identity?

Nov 2, 2014

Pythagorean Identity

${\cos}^{2} \theta + {\sin}^{2} \theta = 1$

I hope that this was helpful.

Jan 8, 2016

The Pythagorean identity is:

color(red)(sin^2x+cos^2x=1

However, it does not have to apply to just sine and cosine.

To find the form of the Pythagorean identity with the other trigonometric identities, divide the original identity by sine and cosine.

SINE:

$\frac{{\sin}^{2} x + {\cos}^{2} x = 1}{\sin} ^ 2 x$

This gives:

${\sin}^{2} \frac{x}{\sin} ^ 2 x + {\cos}^{2} \frac{x}{\sin} ^ 2 x = \frac{1}{\sin} ^ 2 x$

Which equals

color(red)(1+cot^2x=csc^2x

To find the other identity:

COSINE:

$\frac{{\sin}^{2} x + {\cos}^{2} x = 1}{\cos} ^ 2 x$

This gives:

${\sin}^{2} \frac{x}{\cos} ^ 2 x + {\cos}^{2} \frac{x}{\cos} ^ 2 x = \frac{1}{\cos} ^ 2 x$

Which equals

color(red)(tan^2x+1=sec^2x

These identities can all be algebraically manipulated to prove many things:

$\left\{\begin{matrix}{\sin}^{2} x = 1 - {\cos}^{2} x \\ {\cos}^{2} x = 1 - {\sin}^{2} x\end{matrix}\right.$

$\left\{\begin{matrix}{\tan}^{2} x = {\sec}^{2} x - 1 \\ {\cot}^{2} x = {\csc}^{2} x - 1\end{matrix}\right.$