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7.2 Simplifying Trigonometric Expressions
In this section we are going to be combining everything you know about the definitions
of the six trig functions and the Pythagorean Identities presented in the previous lesson.
Example Simplify cscθ tanθ.
To do this, you always want to see if the trig functions given can be expressed in terms of
sines and cosines. The reason why you want to do this is because tangent, cotangent,
secant, and cosecant can all be rewritten in terms of sines and/or cosines.
cscθ = 1/sinθ
cscθ tanθ =
tanθ = sinθ/cosθ
1 sin 
1

 sec 
=
sin  cos  cos 
Example Simplify
1  sin 2 x
cot x
Sometimes you will have to manipulate a Pythagorean Identity to assist you in
simplifying a trig expression.
(cos x)2 + (sin x)2 = 1
(cos x)2 = 1 – (sin x)2
sin x
sin x
1  sin 2 x
cos 2 x
=
= cos 2 x  cot x = cos 2 x  tan x = cos 2 x 
= cos x cos x 
cos x
cos x
cot x
cot x
= cos x sin x
Try the following:
Simplify.
1. sec x (cos x – cos3x)
2. cot x cos x
3. sec2 x (sin x + cos x)2
Answers: sin2x ; csc x ; sec2 x + 2tan x
Example Factor cos 2 x 
cos 2 x 
2
1
sec x
2
 1 = cos 2 x  2cos x  1 = (cos x – 1)(cos x – 1) = (cos x – 1)2
sec x
Try the following:
Factor and simplify.
1. cot2 x – cos2 x
2.
sin 2 x sin 2 x

cot x cos 2 x
Answers: cos2 x cot2 x ; -sin2 x