The magic identity 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
Example. Simplify the expression
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
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Mohamed A. Khamsi