Section 6.3
Proving Identities
Name: ________________
Date: ________________
Verifying vs Proving
Verifying an identity using a specific value validates that it is true for that _______ only. Proving an identity is
done ____________ and validates the identity for all ___________ values of the variable.
To prove that an identity is true for all permissible values, it is necessary to express ____________________
in equivalent forms. One or both sides of the identity must be ______________________ into an equivalent
form to match the other side.
You cannot perform operations ______________________ when proving a potential identity. Simplify the
expressions on each side of the identity ___________________.
Example 1: Verifying Versus Proving That an Equation Is an Identity
Consider the following equation:
1 − sin2 x = sin x cos x cot x
 Determine the non-permissible values for x.
First, determine the non-permissible values.
The only function in the equation that has non-permissible values in its domain is __________.
Recall that cot x is undefined when ____________.
Therefore, x ≠ 180°n, where n ∊ I.
Verify the identity graphically and numerically.
 Verify that the equation is true for some values of x.
Method 1: Verify Graphically
Use technology to graph y = 1 − sin2 x and
y = sin x cos x cot x over the domain −360° ≤ x ≤ 360°.
The graphs appear to be the same. So, graphically, it
seems that 1 − sin2 x = sin x cos x cot x is an identity.
Method 2: Verify Numerically
Use x = 300
 Prove that 1 − sin2 x = sin x cos x cot x for all permissible values of x.
Example 2: Prove an Identity Using Double-Angle Identities
Consider the equation:
tan 𝑥 =
1−cos 2𝑥
sin 2𝑥
Prove that it is an identity for all permissible values of x.
Therefore, tan 𝑥 =
1−cos 2𝑥
sin 2𝑥
is an identity for all permissible values of x.
sin 𝑥
Note: in this example, we could have converted tanx to cos 𝑥 which would have saved one step on the left
side. Sometimes it is advisable to convert all trigonometric functions to expressions of sine or cosine.
Example 3: Prove More Complicated Identities
Prove that the following equation is an identity for all permissible values of x.
1 − cos 𝑥
sin 𝑥
=
sin 𝑥
1 + cos 𝑥
Example 4: Prove an Identity That Requires Factoring
Prove that the following equation is an identity for all permissible values of x.
cos 2𝑥 − cos 𝑥
cot 𝑥 − csc 𝑥 =
sin 2𝑥 + sin 𝑥
Therefore, cot 𝑥 − csc 𝑥 =
cos 2𝑥−cos 𝑥
sin 2𝑥+sin 𝑥
is an identity for all permissible values of x.
Helpful Hints/Rules of Thumbs
 It is usually easier to make a complicated expression simpler than it is to make a simple expression more
complicated (especially when deciding which side of your equation to make like the other).
 Some strategies that may help you prove identities include:
o Use known identities to make substitutions.
o If quadratics are present, the Pythagorean identity or one of its alternate forms can often be used.
o Rewrite the expression using sine and cosine only.
o Multiply the numerator and the denominator by the conjugate of an expression.
o Factor to simplify expressions.
Homework pg 208 #’s 1-4 odd letters; 6; 7a,c; 8a
Reciprocal Identities
csc 𝑥 =
1
sin 𝑥
sec 𝑥 =
1
cos 𝑥
cot 𝑥 =
1
tan 𝑥
Quotient Identities
tan 𝑥 =
sin 𝑥
cos 𝑥
cot x =
cos 𝑥
sin 𝑥
Forms of the Pythagorean Identity
cos2 θ + sin2 θ = 1
cot 2 θ + 1 = csc 2 θ
1 + tan2 θ = sec 2 θ
Sum Identities
Difference Identities
sin(𝐴 + 𝐵) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
cos(𝐴 − 𝐵) = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵
tan(𝐴 + 𝐵) =
tan 𝐴 + tan 𝐵
1 − tan 𝐴 tan 𝐵
tan(𝐴 − 𝐵) =
tan 𝐴 − tan 𝐵
1 + tan 𝐴 tan 𝐵
Double-Angle Identities
sin 2𝐴 = 2 sin 𝐴 cos 𝐴
cos 2𝐴 = cos 2 𝐴 − sin2 𝐴
tan 2𝐴 =
cos 2𝐴 = 2cos2 𝐴 − 1
2 tan 𝐴
1 − tan2 𝐴
cos 2𝐴 = 1 − 2sin2 𝐴
Special Angles and Their Exact Trigonometric Values
Degrees
Radians
sin 𝜃
cos 𝜃
tan 𝜃
1
30°
𝜋
6
1
2
√3
2
45°
𝜋
4
√2
2
√2
2
1
60°
𝜋
3
√3
2
1
2
√3
√3