Showing that two sides of a potential trigonometric identity are
equal for a given value is not enough proof that it is true for all
permissible values of the variable.
Substitute for familiar trig relationships
Express in terms of sine or cosine.
If the expression contains squared terms or 1, -1, try using
a Pythagorean Identity.
Multiply, expand, factor, reduce or square.
Common denominator to add or subtract.
Multiply by the conjugate of a binomial.
Math 30-1
1
Proving an Equation is an Identity
2
sin
A 1
1
Consider the equation
1
.
2
sin A  sinA
sinA
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)
sin2 A  1
y
sin2 A  sin A
1
y 1
sin A
Math 30-1
2
Proving an Equation is an Identity [Cont’d]
b) Verify that this statement is true for x = 2.4 rad.
sin2 A  1
sin2 A  sinA

1
1
sinA
(sin 2.4)2  1

(sin 2.4)2  sin2.4
1
 1
sin 2.4
= 2.480466
= 2.480466
L.S. = R.S.
Therefore, the equation is true for x = 2.4 rad.
Math 30-1
3
Proving an Equation is an Identity
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
Note the left side of the
sin2 A  1
sin2 A  sinA

1
1
sinA
(sinA  1)(sinA  1)

sinA(sinA  1)
1
1
sin A
(sin A  1)

sin A
sinA
1


sinA sin A
1
 1
sin A
L.S. = R.S.
Math 30-1
equation has the restriction
sin2A - sin A ≠ 0
sin A(sin A - 1) ≠ 0
sin A ≠ 0 or sin A ≠ 1

A  0,  or A 
2
Therefore,A  0  2  n or
A   + 2 n, or

A   2  n, wheren is
2
any integer.
The right side of the
equation has the restriction
sin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0,  + 2 n,
4
where n is any integer.
Proving an Identity
common denominator
1
1

 2 csc 2 x
1  cos x 1  cos x
(1  cos x)  (1  cos x) 2 csc 2 x
(1  cos x)(1  cos x)
2
(1  cos 2 x)
2
sin 2 x
2 csc 2 x
L.S. = R.S.
x  n , n  
x  180 n, n  
Math 30-1
5
Proving an Identity
factoring
sin4x - cos4x
=
1 - 2cos2 x
= (sin2x - cos2x)(sin2x + cos2x) 1 - 2cos2x
= (sin2x - cos2x)(1)
= 1 - cos2x - cos2x
= 1 - 2cos2x
L.S. = R.S.
Math 30-1
6
Proving an Equation is an Identity Substitution
sin 2
cos 2  1
2sin  cos 
cos 2   sin 2   1

tan
sin 
cos 
2sin  cos 
cos 2   1  sin 2 
2sin  cos 
2 cos 2 
sin 
cos 
30-1
L.S. =Math
R.S.
7
Proving an Equation is an Identity Multiplying by Conjugates
1  cos 
sin 

sin 
1  cos
sin 

1  cos 
1  cos 

1  cos 
sin  1  cos  

1  cos 2 
sin  1  cos  

sin 2 
1  cos 

sin 
Math 30-1
8
Textbook p. 314 – 315
Low: (Basic Drill and Practice)
1 – 7, 8
Medium: (Problem Solving and Word Problems)
9, 19
High: (Extension and Higher Level)
10 – 18, C2, C3
Math 30-1
9