15.2 VERIFYING TRIG IDENTITIES

Verifying trig identities algebraically involves transforming one side of the equation into the same form as the other side using basic trig identities and properties of algebra.

Procedure for Verifying Trig Identities

1. Draw a vertical line below the equal sign in the equation.

2. Determine which side of the identity to change (usually the more complicated side)

3. Use fundamental trig identities and algebraic properties to change the chosen side into the other.

Suggestions/Strategies for Verifying

Start with the more complicated side – you are trying to get it to match the simpler side

Get a common denominator Substitute one or more basic trig functions

i.e. you see a 1 and you can replace it with Factor or multiply to simplify expressions

i.e. Multiply expression by an expression of 1

Remember the fancy form of 1, or

Express all trig functions in terms of sine and cosine i.e. you see and you replace it with

sin2 x + cos2 x

sin2 x−cos2 x= sinx−cosx( ) sinx+ cosx( )

same#

same#

same expression

same expression

1

cos x

Remember, there are many different ways to transform the equations so there isn’t necessarily one right way to simplify!

Review of the Pythagorean Identities

Write four different ways:

sin2θ + cos2θ =1

Write two different ways

tan2θ +1=sec2θ

Write two different ways

cot2θ +1=csc2θ

Verify the identity.

1. csc x−sinx=cosxcotx

Verify the identity.

2. cos x + sinxtanx=secx

Verify the identity.

3. sec x csc x−cotxcosx( ) =tanx

Verify the identity.

4. sec x csc x−sinx( ) =cotx

Verify the identity.

5. tan x + cotx( )2 =csc2 xsec2 x

Verify the identity.

6. csc x−sinx( )2 =cot2 x−cos2 x