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MATH 201 - Week 10
Ferenc Balogh
Concordia University
2008 Winter
Based on the textbook
J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics for Calculus, 5th Edition, Thomson
All figures and videos are made using MAPLE 11 and ImageMagick-convert.
Overview
Trigonometric Identities - Section 7.1
Trigonometric Identity vs Trigonometric Equation
The Starting Point: Fundamental Trigonometric Identities
Simplifying Trigonometric Expressions
Proving Trigonometric Identities
Trigonometric Identity
A trigonometric identity is an equation of trigonometric expressions
in θ valid for all values of θ.
For example
sin2 θ + cos2 θ = 1
is valid for all θ.
Trigonometric Equation
A trigonometric equation is an equation of trigonometric
expressions in θ valid for some particular values of θ.
For example
sin θ + cos θ = 1
is valid for θ = 0 but it is invalid for θ = π4 .
Fundamental Trigonometric Identities
Reciprocal Identities
csc x =
1
sin x
sec x =
tan x =
sin x
cos x
1
cos x
cot x =
cot x =
1
tan x
cos x
sin x
Pythagorean Identities
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Even-Odd Identities
sin(−x) = − sin x
cos(−x) = cos x
tan(−x) = − tan x
Cofunction Identities
sin
π
− x = cos x
2
π
cos
− x = sin x
2
tan
cot
π
2
π
2
− x = cot x
− x = tan x
sec
csc
π
2
π
2
− x = csc x
− x = sec x
Although these are considered to be fundamental identities, it is
more convenient to prove them later on.
Example. Simplify the trigonometric expression
sin t + cos t cot t.
Solution.
cos t
sin t
2
2
sin t + cos t
sin t
1
sin t
sin t + cos t cot t = sin t + cos t
=
=
= csc t.
Example. Simplify the trigonometric expression
cos x
sin x
+
.
sin x
1 + cos x
Solution.
sin x
cos x
+
sin x
1 + cos x
=
=
=
=
cos x(1 + cos x) + sin2 x
sin x(1 + cos x)
cos x + cos2 x + sin2 x
sin x(1 + cos x)
cos x + 1
sin x(1 + cos x)
1
sin x
= csc x.
How to prove a trigonometric identity?
We transform one side of the equation into the other side by using
a sequence of steps.
Hints
I
Start with one side (take the more complicated one).
I
Use the fundamental identities and perform algebraic
manipulations.
I
In case of an emergency, write all expressions in terms of sines
and cosines only.
I
PRACTICE!!!
Example. Prove the identity
sec x − cos x
= sin2 x.
sec x
rewriting it in terms of sines and cosines.
Solution. The LHS is
LHS
=
=
=
sec x − cos x
sec x
1
cos x − cos x
1
cos x
1−cos2 x
cos x
1
cos x
2
=
1 − cos x cos x
·
cos x
1
1 − cos2 x
=
sin2 x
=
= RHS.
Example. Verify the identity
1
1
+
= 2 sec x.
sec x + tan x
sec x − tan x
Solution.
LHS
=
=
=
=
=
1
1
+
sec x + tan x
sec x − tan x
sec x − tan x + sec x + tan x
(sec x + tan x)(sec x − tan x)
2 sec x
sec2 x − tan2 x
2 sec x
1
2 sec x
= RHS.
Example. Verify the identity
sin4 t − cos4 t = sin2 t − cos2 t.
Solution.
LHS
= sin4 t − cos4 t
= sin4 t − (cos2 t)2
= sin4 t − (1 − sin2 t)2
= sin4 t − (1 − 2 sin2 t + sin4 t)
= sin4 t − 1 + 2 sin2 t − sin4 t
= −1 + 2 sin2 t
= −(sin2 t + cos2 t) + 2 sin2 t
= − sin2 t − cos2 t + 2 sin2 t
= sin2 t − cos2 t
= RHS.
Example. Verify the identity
1 + tan2 u
1
.
=
2
2
1 − tan u
cos u − sin2 u
Solution.
1 + tan2 u
LHS =
1 − tan2 u
=
=
=
=
=
sin u 2
cos u
sin u 2
1 − cos
u
cos2 u+sin2 u
cos2 u
cos2 u−sin2 u
cos2 u
cos2 u + sin2 u
1+
·
cos2 u
cos2 u − sin2 u
cos2 u
+ sin2 u
cos2 u − sin2 u
1
= RHS.
2
cos u − sin2 u
cos2 u
Example. Verify the identity
tan v − sin v
tan v sin v
=
tan v + sin v
tan v sin v
Solution.
LHS =
tan v sin v
tan v + sin v
=
=
=
=
=
sin v
cos v sin v
sin v
cos v + sin v
sin2 v
cos v
sin v +sin v cos v
cos v
2
sin v
cos v
·
cos v sin v + sin v cos v
sin2 v
sin v + sin v cos v
sin v
1 + cos v
Solution (cont).
RHS =
tan v − sin v
tan v sin v
=
=
=
=
=
=
=
sin v
cos v − sin v
sin v
cos v sin v
sin v −sin v cos v
cos v
sin v cos v
cos v
sin v − sin v cos v cos v
· 2
cos v
sin v
1 − cos v
sin v
1 − cos v 1 + cos v
·
sin v
1 + cos v
2
1 − cos v
sin v (1 + cos v )
sin2 v
sin v
=
.
sin v (1 + cos v )
1 + cos v