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

and

Of course, we used the fact that

which gives

But

and since

and , we get finally

or we may take . In this case we have

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

Since

we get

Since

we get

Finally we have

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

**addition formulas**and

Of course, we used the fact that

**Example.**verify the identity**Answer.**We havewhich 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 getor we may take . In this case we have

**Example.**Find the exact value of**Answer.**We haveHence, using the additions formulas for the cosine function we get

Since

we get

**Example.**Find the exact value for**Answer.**We haveSince

we get

Finally we have

**Remark.**Using the addition formulas, we generate the following identitiesMore identities may be proved similar to the above ones. The bottom line is to remember the addition formulas and use them whenever needed.