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 take

*a*=

*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