Questions 1:
Simplify the following trigonometric expression.
csc (x) sin (Pi/2 - x)
Solution to Question 1:
-
Use the identity sin (Pi/2 - x) = cos(x) and simplify
csc (x) sin (Pi/2 - x)= csc (x) cos (x) = cot (x)
Questions 2:
Simplify the following trigonometric expression.
[sin 4x - cos 4x] / [sin 2x - cos 2x]
Solution to Question 2:
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Factor the denominator
[sin 4x - cos 4x] / [sin 2x - cos 2x]
= [sin 2x - cos 2x][sin 2x + cos 2x] / [sin 2x - cos 2x]
-
and simplify
= [sin 2x + cos 2x] = 1
Questions 3:
Simplify the following trigonometric expression.
[sec(x) sin 2x] / [1 + sec(x)]
Solution to Question 3:
-
Substitute sec (x) that is in the numerator by 1 / cos (x) and simplify.
[sec(x) sin 2x] / [1 + sec(x)]
= sin 2x / [ cos x (1 + sec (x) ]
= sin 2x / [ cos x + 1 ]
-
Substitute sin 2x by 1 - cos 2x , factor and simplify.
= [ 1 - cos 2x ] / [ cos x + 1 ]
= [ (1 - cos x)(1 + cos x) ] / [ cos x + 1 ] = 1 - cos x
Questions 4:
Simplify the following trigonometric expression.
sin (-x) cos (Pi / 2 - x)
Solution to Question 4:
-
Use the identities sin (-x) = - sin (x) and cos (Pi / 2 - x) = sin (x) and simplify
sin (-x) cos (Pi / 2 - x) = - sin (x) sin (x) = - sin 2x
Questions 5:
Simplify the following trigonometric expression.
sin 2x - cos 2x sin 2x
Solution to Question 5:
-
Factor sin 2x out, group and simplify
sin 2x - cos 2x sin 2x
= sin 2x ( 1 - cos 2x )
= sin 4x
Questions 6:
Simplify the following trigonometric expression.
tan 4x + 2 tan 2x + 1
Solution to Question 6:
-
Note that the given trigonometric expression can be written as a square
tan 4x + 2 tan 2x + 1
= ( tan 2x + 1) 2
-
We now use the identity 1 + tan 2x = sec 2x
= ( sec 2x ) 2 = sec 4x
Questions 7:
Add and simplify.
1 / [1 + cos x] + 1 / [1 - cos x]
Solution to Question 7:
-
In order to add the fractional trigonometric expressions, we need to have a common denominator
1 / [1 + cos x] + 1 / [1 - cos x]
= [ 1 - cos x + 1 + cos x ] / [ [1 + cos x] [1 - cos x] ]
= 2 / [1 - cos 2x]
= 2 / sin 2x = 2 csc 2x
Questions 8:
Write sqrt( 4 - 4 sin 2x ) without square root for Pi / 2 < x < Pi.
Solution to Question 8:
-
Factor, and substitute 1 - sin 2x by cos 2x
sqrt( 4 - 4 sin 2x )
= sqrt[ 4(1 - sin 2x ) ]
= 2 sqrt[ cos 2x ]
= 2 | cos (x) |
-
Since Pi / 2 < x < Pi, cos x is less than zero and the given trigonometric expression simplifies to
= - 2 cos (x)
Questions 9:
Simplify the following expression.
[1 - sin 4x] / [1 + sin 2x]
Solution to Question 9:
-
Factor the denominator, and simplify
[1 - sin 4x] / [1 + sin 2x]
= [1 - sin 2x] [1 + sin 2x] / [1 + sin 2x]
= [1 - sin 2x] = cos 2x
Questions 10:
Add and simplify.
1 / [1 + sin x] + 1 / [1 - sin x]
Solution to Question 10:
-
Use a common denominator to add
1 / [1 + sin x] + 1 / [1 - sin x]
= [1 - sin x + 1 + sin x] / [ (1 + sin x)(1 - sin x) ]
= 2 / [ 1 - sin 2x ]
= 2 / cos 2x = 2 sec 2x
Questions 11:
Add and simplify.
cos x - cos x sin 2x
Solution to Question 11:
-
factor cos x out
cos x - cos x sin 2x
= cos x (1 - sin 2x)
= cos x cos 2x = cos 3x
Questions 12:
Simplify the following expression.
tan 2x cos 2x + cot 2x sin 2x
Solution to Question 12:
-
Use the trigonometric identities tan x = sin x / cos x and cot x = cos x / sin x to write the given expression as
tan 2x cos 2x + cot 2x sin 2x
= (sin x / cos x) 2 cos 2x + (cos x / sin x) 2 sin 2x
-
and simplify
= sin 2x + cos 2x = 1
Questions 13:
Simplify the following expression.
sec (Pi/2 - x) - tan(Pi/2 - x) sin(Pi/2 - x)
Solution to Question 13:
-
Use the identities sec (Pi/2 - x) = csc x, tan(Pi/2 - x) = cot x and sin(Pi/2 - x) = cos x to write the given expression as
sec (Pi/2 - x) - tan(Pi/2 - x) sin(Pi/2 - x)
= csc x - cot x cos x = csc x - (cos x / sin x) cos x
= csc x - cos 2x / sin x
= 1 / sin x - cos 2x / sin x
= (1 - cos 2x) / sin x
= sin 2x / sin x
= sin x
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