# Pythagorean Identities

Let's look at how it works. Given the unit circle, which has a radius of 1, and any point on the circle that creates the vertex of a right triangle can be represented by the coordinates (x, y)

Recall the Pythagorean Theorem is x

^{2}+ y

^{2}= c

^{2}. Since the legs of the right triangle can be represented by sin θ and cos θ and the radius is the hypotenuse we can use the Pythagorean Theorem to derive sin

^{2}θ + cos

^{2}θ = 1.

From this fundamental identity we can derive

*equivalent identities*such as:

- sin
^{2}θ + cos^{2}θ = 1 - sin
^{2}θ + cos^{2}θ = 1 - using the division property of equality

-cos

^{2}θ -cos

^{2}θ using the subtraction property of equality

sin^{2} θ = 1 - cos^{2} θ |

-sin

^{2}θ -sin

^{2}θ using the subtraction property of equality

cos^{2} θ = 1 - sin^{2} θ |

1 + cot

^{2}θ = csc

^{2}θ use substitution

-1-1use subtraction property of equality

cot^{2} θ = csc^{2} θ - 1 |

Deriving the Pythagorean identities leads to understanding the basic trigonometric proofs. Recall that when solving an equation both sides of the equation may be manipulated to find a solution. However, when proving an identity only one side of the identity may be manipulated.

Related Links:Math Trigonometry Cofunction Identities Sum and Difference of Angles Identities |

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