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Verifying
Trigonometric
Identities
Basic Trigonometric Identities
Reciprocal Identities
1
csc x = sin x
1
sec x = cos x
1
cot x = tan x
Quotient Identities
sin x
tan x = cos x
cos x
cot x = sin x
Pythagorean Identities
sin2x + cos2x = 1
tan2x + 1 = sec2x
1 + cot2x = csc2x
Even-Odd Identities
sin(-x) = -sin x
csc(-x) = -csc x
cos(-x) = cos x
sec(-x) = sec x
tan(-x) = -tan x
cot(-x) = -cot x
Text Example
Verify the identity: sec x cot x = csc x.
Solution
The left side of the equation contains the more complicated
expression. Thus, we work with the left side. Let us express this side of the
identity in terms of sines and cosines. Perhaps this strategy will enable us to
transform the left side into csc x, the expression on the right.
1
cos x
sec x cot x =

cos x sin x
Apply a reciprocal identity: sec x = 1/cos x
and a quotient identity: cot x = cos x/sin x.
1
=
= csc x
sin x
Divide both the numerator and the
denominator by cos x, the common factor.
Text Example
Verify the identity: cosx - cosxsin2x = cos3x..
Solution
We start with the more complicated side, the left side. Factor out
the greatest common factor, cos x, from each of the two terms.
cos x - cos x sin2 x = cos x(1 - sin2 x)
= cos x ·
= cos3 x
cos2
x
Factor cos x from the two terms.
Use a variation of sin2 x + cos2 x = 1.
Solving for cos2 x, we obtain cos2 x =
1 – sin2 x.
Multiply.
We worked with the left and arrived at the right side. Thus, the identity is
verified.
Guidelines for Verifying
Trigonometric Identities
1.
Work with each side of the equation independently of the other
side. Start with the more complicated side and transform it in a
step-by-step fashion until it looks exactly like the other side.
2.
Analyze the identity and look for opportunities to apply the
fundamental identities. Rewriting the more complicated side
of the equation in terms of sines and cosines is often helpful.
3.
If sums or differences of fractions appear on one side, use the
least common denominator and combine the fractions.
4.
Don't be afraid to stop and start over again if you are not
getting anywhere. Creative puzzle solvers know that
strategies leading to dead ends often provide good problemsolving ideas.
Example
• Verify the identity:
csc(x) / cot (x) = sec (x)
Solution:
csc x
= sec x
cot x
1
sin x = 1
cos x cos x
sin x
1
sin x
1

=
sin x cos x cos x
Example
• Verify the identity:
cos x = cos x  cos x sin x
3
2
Solution:
cos x = cos x  cos x sin x
3
2
cos x = cos x(cos x  sin x)
cos x = cos x(1)
2
2
Example
• Verify the following identity:
tan 2 x - cot 2 x
= tan x - cot x
tan x  cot x
Solution:
sin 2 x cos 2 x
tan 2 x - cot 2 x cos 2 x sin 2 x
=
sin x cos x
tan x  cot x

cos x sin x
sin 4 x - cos 4 x
4
4
2
2
sin
x
cos
x cos x sin x
cos
x
sin
x
=
=

2
2
2
2
sin x  cos x
cos x sin x
1
cos x sin x
Example cont.
Solution:
sin 4 x - cos 4 x
cos x sin x
(sin 2 x  cos 2 x)(sin 2 x - cos 2 x)
=
sin x cos x
sin 2 x - cos 2 x
sin 2 x
cos 2 x
=
=
sin x cos x
sin x cos x sin x cos x
sin x cos x
=
= tan x - cot x
cos x sin x
sin 4 x - cos 4 x = 1 - 2 cos 2 x
cot
2


t  1 sin t  1 = cot t  2
2
2