Pythagorean identities

The Pythagorean identities state that

$\sin^2x + \cos^2x = 1$

$1 + \cot^2x = \csc^2x$

$\tan^2x + 1 = \sec^2x$

Using the unit circle definition of trigonometry, because the point $(\cos (x), \sin (x))$ is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, $\sin^2x + \cos^2x = 1$.

Another way to think of it is as follows: Suppose that there is a right triangle $ABC$ with the right angle at $B$. Then, we have:




To derive the other two Pythagorean identities, divide $\sin^2A+cos^2A+1$ by either $\sin^2 (x)$ or $\cos^2 (x)$ and substitute the respective trigonometry in place of the ratios to obtain the desired result.

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