1. Trigonometric
Identities

2. Prove that Trigonometric Identities
A trigonometric equation is an equation that involves
at least one trigonometric function of a variable. The
equation is a trigonometric identity if it is true for all
values of the variable for which both sides of the
equation are defined.
Prove that
Recall the basic
trig identities:
L.S. = R.S.

3. Trigonometric Identities
Quotient Identities
Reciprocal Identities
Pythagorean Identities
sin2q + cos2q = 1
tan2q + 1 = sec2q
cot2q + 1 = csc2q
sin2q = 1 - cos2q
tan2q = sec2q - 1
cot2q = csc2q - 1
cos2q = 1 - sin2q

4. Trigonometric Identities [cont’d]
sinx x sinx = sin2x
= cosA

5. Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify. b) a)

6. Simplifying Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x d)

7. Proving an Identity
Steps in Proving Identities
1. Start with the more complex side of the identity and work
with it exclusively to transform the expression into the
simpler side of the identity.
2. Look for algebraic simplifications:
Do any multiplying , factoring, or squaring which is
obvious in the expression.
Reduce two terms to one, either add two terms or
factor so that you may reduce.
3. Look for trigonometric simplifications:
Look for familiar trig relationships.
If the expression contains squared terms, think
of the Pythagorean Identities.
Transform each term to sine or cosine, if the
expression cannot be simplified easily using other ratios.
4. Keep the simpler side of the identity in mind.

8. Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x
= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x
c) tan x sin x + cos x = sec x
L.S. = R.S.
L.S. = R.S.

9. Proving an Identity
d) sin4x - cos4x = cos2 x
= (sin2x - cos2x)(sin2x + cos2x)
= (1 - cos2x - cos2x)
= 1 - 2cos2x
1 - 2cos2x
L.S. = R.S. e)
L.S. = R.S.

10. Proving an Identity f)
L.S. = R.S.

11. Using Exact Values to Prove an Identity
Consider
a) Use a graph to verify that the equation is an identity.
b) Verify that this statement is true for x =
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions. a)

12. Using Exact Values to Prove an Identity [cont’d]
b) Verify that this statement is true for x =
Rationalize the
denominator:
Therefore, the identity is
true for the particular
case of
L.S. = R.S.

13. Using Exact Values to Prove an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
Restrictions:
Note the left side of the
equation has the restriction
1 - cos x ≠ 0 or cos x ≠ 1.
Therefore, x ≠ 0 + 2p n,
where n is any integer.
The right side of the
equation has the restriction
sin x ≠ 0. x = 0 and p
Therefore, x ≠ 0 + 2pn
and x ≠ p + 2pn, where
n is any integer.
L.S. = R.S.

14. Proving an Equation is an Identity
Consider the equation
a) Use a graph to verify that the equation is an identity.
b) Verify that this statement is true for x = 2.4 rad.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions. a)

15. Proving an Equation is an Identity [cont’d]
b) Verify that this statement is true for x = 2.4 rad. = =
L.S. = R.S.
Therefore, the equation is true for x = 2.4 rad.

16. Proving an Equation is an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
Note the left side of the
equation has the restriction:
sin2A - sin A ≠ 0
sin A(sin A - 1) ≠ 0
sin A ≠ 0 or sin A ≠ 1
The right side of the
equation has the restriction
sin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0, p + 2p n,
where n is any integer.
L.S. = R.S.