3) The Addition Formulas

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

Answer. We have

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 takea=b = 0, we get

or we may take  . In this case we have


Example. Find the exact value of

Answer. We have

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

Since

we get


Example. Find the exact value for


Answer. We have

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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