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An identity, in mathematics, is an equation that is true for all possible values.
(It is always true regardless of the values that are substituted into the equation.)
A trigonometric identity is an equation that involves trigonometric functions and is true for every single value substituted for the variable (assuming both sides are "defined" for that value) You will find that trigonometric identities are especially useful for simplifying trigonometric expressions.

The most common trigonometric identities are those involving the Pythagorean Theorem.

When studying the unit circle (radius of 1), it was observed that a point on the unit circle (a vertex of the right triangle) can be represented by the coordinates (cos θ, sin θ ).

Since the legs of the right triangle in the unit circle have the values of sin θ and cos θ, the Pythagorean Theorem can be used to obtain sin2 θ + cos2 θ = 1.

This well-known equation is called a Pythagorean Identity
It is true for all values of θ in the unit circle.

Using this first Pythagorean Identity, two additional Pythagorean Identities can be created.
 • Start with this first Pythagorean Identity. • Divide each term by cos2θ. • We know and . • Substitute and simplify.

We now have a second Pythagorean Identity:

It should be noted that there are values of θ for which tangent and secant are undefined.

If we divide by a different value, we can arrive at the third identity:
 • Start with this first Pythagorean Identity. • Divide each term by sin2θ. • We know and . • Substitute and simplify.

The third Pythagorean Identity is:

 When working with Pythagorean Identities, keep in mind that there are other ways to express (rearrange) these identities that may be useful.
 Pythagorean Identity Other Variation(s)

 Use Pythagorean Identities to find missing trigonometric values.

find the value of the other trig functions.
 Yes, this problem can also be solved without the use of a Pythagorean Identity.

Solution:
The easiest Pythagorean Identity to work with is
sin2θ +cos2θ = 1.

 Use Pythagorean Identities to help simplify trig expressions.

Simplify:    sinx cos2x - sinx

Solution: Since this expression contains sine and cosine, utilize sin2θ +cos2θ = 1 or its variations.
Start by factoring:
 Use this substitution: sin2x +cos2x = 1 gives cos2x - 1 = -sin2x

 Use Pythagorean Identities to help simplify a trig expression into a factorable form.

Express csc2x - cotx - 3 in factored form.

Solution:
Start by substituting:
 Use this Pythagorean Identity: 1 + cot2x = csc2x

You will find that these Pythagorean Identities
will also be valuable when solving trigonometric equations.

 For help with verifying trig identities on your calculator, click here.

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