Pythagorean Identities

Consider a right triangle with a hypotenuse of 1.

By recalling the sine and cosine ratios, the lengths of the opposite and adjacent sides to ∠ θ can be expressed in terms of the angle.

	Definition	Substitute	Simplify
sin θ	Length of oppositeside to∠ θ/Hypotenuse	opp/1	opp
cos θ	Length of adjacentside to∠ θ/Hypotenuse	adj/1	adj

It can be seen that if the hypotenuse of a right triangle is 1, the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.

By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of sin θ and cos θ is equal to the square of 1.

Since cos θ represents a side length, it is not 0. Therefore, by diving both sides of the above equation by cos^2θ, the second identity can be obtained.

The second identity was obtained.

Since sin θ represents a side length, it is not 0. Therefore, by dividing both sides of sin ^2 θ +cos ^2 θ = 1 by sin ^2 θ, the third identity can be proven.

Finally, the third identity was obtained.

The first identity can be shown using the unit circle and the Pythagorean Theorem. Consider a point (x,y) on the unit circle in the first quadrant, corresponding to the angle θ. A right triangle can be constructed with θ.

By the Pythagorean Theorem, the sum of the squares of x and y equals 1. x^2 + y^2 = 1 In fact, this is true not only for points in the first quadrant, but for every point on the unit circle. Recall that, for points (x,y) on the unit circle corresponding to angle θ, it is known that x = cos θ and that y = sin θ. By substituting these expressions into the equation, the first identity can be obtained.

Dividing both sides by either cos^2 θ or sin^2 θ leads to two variations of the Pythagorean Identity.