## ** The magic identity**

Trigonometry is the art of doing algebra over the circle. So it is a mixture of algebra and geometry. The sine and cosine functions are just the coordinates of a point on the unit circle. This implies the most fundamental formula in trigonometry (which we will call here the **magic identity**)

where is any real number (of course measures an angle).

**Example.** Show that

**Answer.** By definitions of the trigonometric functions we have

Hence we have

Using the magic identity we get

This completes our proof.

**Remark.** the above formula is fundamental in many ways. For example, it is very useful in techniques of integration.

**Example.** Simplify the expression

**Answer.** We have by definition of the trigonometric functions

Hence

Using the magic identity we get

Putting stuff together we get

This gives

Using the magic identity we get

Therefore we have

**Example.** Check that

**Answer.**

**Example.** Simplify the expression

**Answer.**

The following identities are very basic to the analysis of
trigonometric expressions and functions. These are called **
Fundamental Identities**

**Reciprocal identities**

**Pythagorean Identities**

**Quotient Identities**

**
**

**
[Trigonometry]
[Addition Formulas]
**** **
[Geometry]
[Algebra]
[Differential Equations]
[Calculus]
[Complex Variables]
[Matrix Algebra]

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*Mohamed A. Khamsi *

Tue Dec 3 17:39:00 MST 1996

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