The fundamental identities are very important for the analysis of trigonometric expressions and functions but they are a direct result of the intimate relation between trigonometry and geometry. The power behind the algebraic nature of trigonometry is hidden and can be measured only with the addition formulas

and

Of course, we used the fact that

Example. verify the identity

which gives

But

and since

and , we get finally

Remark. In general it is good to check whether the given formula is correct. One way to do that is to substitute some numbers for the variables. For example, if we take a=b = 0, we get

or we may take . In this case we have

Example. Find the exact value of

Hence, using the additions formulas for the cosine function we get

Since

we get

Example. Find the exact value for

Since

we get

Finally we have

Remark. Using the addition formulas, we generate the following identities

More identities may be proved similar to the above ones. The bottom line is to remember the addition formulas and use them whenever needed.

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[Calculus] [Complex Variables] [Matrix Algebra]