Focus on…

Proving trigonometric identities algebraically

Understanding the difference between verifying and proving an identity

Showing that verifying that the two sides of a potential identity are equal for a given value is insufficient to prove the identity
To prove that an identity is true for all permissible values, it is necessary to express both sides of the identity in equivalent forms. One or both sides of the identity must be algebraically manipulated into an equivalent form to match the other side.
You cannot perform operations across the equal sign when proving a potential identity. Simplify the expressions on each side of the identity independently.
Example 1: Verify Versus Prove That an Equation is an Identity
A) Determine the non-permissible values for the equation tan x cos x

1
 cos
2 x csc x
B) Verify that the equation may be an identity, either graphically using technology or by choosing on value for x .
C) Prove that the identity is true for all permissible values of x .
A) The functions in the equation that have non-permissible values in its domain is tan x and
Recall that tan x is undefined when cos x

0 .
Therefore, x

90
 
180
 n , where n

I .
Recall that x
Therefore, x

180
 n , where n

I . sin x

0 .
B) Verify Numerically
Use x

60

Left side

 tan tan x cos x csc
60
 x cos 60

  csc
1
2
60




2


3
Right side

1
 cos
2

1
 cos
2
60


1

1
2
2 x


3
4
2
3

2
3

1

1
4

3
4

C) To prove the identity algebraically, examine both sides of the equation and simplify each side to a common expression.
Left side
 tan x cos x csc x
Right side

1
 cos
2 x sin x  cos x

 cos x
1
 sin
2 x sin x
 sin x
1
Why is this true?
 sin sin x x
 sin x
 sin
2 x
Left side = Right side
Therefore, tan x cos x

1
 cos
2 x is an identity for x

90
 n , where x

I . csc x
Example 2: Prove an Identity Using Double-Angle Idenitities
Prove that sin cos 2
2 x x

1
 tan x is an identity for all permissible values of x .
Left side
 sin 2 x cos 2 x

1
 cos
2 sin
2 x x cos
 sin
2 x x

1
Right side
 tan x

2 sin
1
 sin
2 x cos x
 sin
2 x x

1

2 sin
2 x cos

2 sin
2 x x

2
2

1 sin
 x cos sin
2 x x

 sin x cos cos
2 x x
 sin x cos x
 tan x
Therefore, sin 2 x cos 2 x

1
Left side = Right side
 tan x is an identity for all permissible values of x .

4.
3.
2.
Exit Card
1. Factor and simplify each rational trigonometric expression.
A) cos x sin sin
2 x x
 cos

1 x
B) cot cos
2 x x

3 cot cot x x

 cos x
4
Use factoring to help to prove each identity for all permissible values of x .
A) cos x sin sin
2 x x
 cos

1 x

1
 sin cos x x
B)
1

2
1
 cos
2 cos x x

3 cos
2 x

1
 cos x
1

3 cos x
Use common denominator to express the rational expressions as a single term.
A) cos
1
 sin x x
 sin x cos x
B) sin csc x x

1
 sin csc x x

1
Prove the identity. cos x
 sin x tan x
 sec x