5.2 Verifying Identities

What is an identity?

Guidelines for Verifying Identities1. Work with one side of the equation at a time. Use the more complicated

side.

2. Try to: factor, add fractions, square a binomial, create a monomial denominator

3. Look to use the fundamental identities—note which functions are in the final expression you want.

4. If all else fails, convert everything to sines and cosines

5. Try something! Even attempts that lead to dead ends provides insight.

**There is no well defined set of rules or procedure to follow. This is best learned by practice.

**Remember, these are NOT equations—you cannot use operations such as adding/subtracting/multiplying/dividing/squaring both sides!

Example 1

1seccos

cossin22

22

Example 2

2csc2cos1

1

cos1

1

Example 3

xxx 222 sin1sin1sec

Example 4

xxxx cotcossincsc

Example 5

u

uuu

cos1

sincotcsc

Example 6

x

x

x

x

cos

cos1

sec1

tan2

Example 7

xxxx tansectantan 23

Example 8 xxxxx cossinsinsincos 6443

What’s More Complicated?1. Fractions with a binomial (or longer) denominator

2. Fractions

3. Addition/Subtraction

4. Multiplication

Some Hints To Try1. See squared terms? Think Pythagorean Identities

2. Don’t see any algebraic steps Think converting to sines and cosines

3. See multiple fractions Combine them

4. See one fraction with a monomial denominator Make it two.

Practice

222 sin21sincos

0coscos

sinsin

sinsin

coscos

yx

yx

yx

yx

1cossinsinsec2sec2 22222 xxxxx

xx

xxcsc

sin

tancos

sin

cos1

cos1

sin

1.

2.

3.

4.

5.

Practice

xxxxxx sincoscossincossin 32

5

2

1

csctansin

sec1

1tantan1tan

1tan 23

xxx

x

xxxx 3235 tansectantan

6.

7.

8.

9.

10. 12

sinsin 22

xx