Section 5.2

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```					1  cot 2  

Warm-up
Complete the identities by filling in the blanks.
1. 1 + cot²θ=_________           2. sin(π/2-θ)=______
3. cos(-x)= __________           4. cos²θ=________

Simplify:
5. (secx -1)(secx +1)

6. Evaluate without using a calculator or reference
triangles.
Find sec θ and csc θ if tan θ =3 and cos θ > 0
Section 5.2
Proving Trigonometric
Identities
To prove an identity means to
show that one side of an
equation equals the other side
of the equation by using
identities or algebraic
manipulations.
Example 1

Prove the identity
sin θ  cos θ
2       2
1
cos θ sec θ
2    2

1. Always work with the more
complicated side of equation.
sin θ  cos θ
2       2
1

cos θ sec θ
2     2
cos θ sec θ
2     2

 1  1 
         sec2 θ 
 cos θ  
2

 1 
    2  
 cos θ 

cos θ
2

1
Example 2 Combining Fractions
Before Using Identities

Prove the identity.
1        1
          2csc β
2

1  cosβ 1  cosβ
Get a common denominator on
the left side of the equation.
1        1
         
1  cosβ 1  cosβ

1 1  cosβ   1 1  cosβ 

1  cosβ 1  cosβ 
1  cosβ  1  cosβ

1  cos β
2
1  cosβ  1  cosβ

1  cos β
2

2

1  cos β
2

2

sin β
2

2csc β
2
Example 3:

Prove the identity.
 sec
2
           
x  1 sin x  1   sin x
2          2

Use identities to simplify the
left side of the equation.
 sec x  1 sin x  1 
2           2

 tan x   cos x  
2           2

 sin x 
2

         
  cos x 
2

 cos x 
2

 sin x
2
Example 4: Change to sines and
cosines

Verify the identity.
csc x  sin x  cos x cot x
When all else fails change one
side to sines and cosines.
1
csc x  sin x         sin x
sin x
1
csc x  sin x         sin x
sin x
1    sin x2
      
sin x sin x
1  sin x
2

sin x
cos x2

sin x
cos x
2

sin x
 cos x 
 cos x        
 sin x 
 cos xcot x
Example 5: Setting up a
Difference of Squares

Prove the identity.
sinβ
cscβ  cotβ 
1  cosβ
Work on the right side of the
equation because you can make a
monomial denominator.
sinβ
cscβ  cotβ 
1  cosβ
sinβ  1  cosβ 
           1  cosβ 
1  cosβ           
sinβ  sinβ cosβ

1  cos β
2
sinβ  sinβ cosβ

1  cos β2

sinβ  sinβ cosβ

sin β
2

sinβ 1  cosβ 

sin β
2
sinβ 1  cosβ 

sin β2

1  cosβ

sinβ
1     cosβ
       
sinβ sinβ
 cscβ  cotβ
Example 6: Working on Both
Sides

Prove the identity.
cot x
2
 (cot x)(sec x  tan x)
1  csc x

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