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 ^{4}x  cos ^{4}x] / [sin ^{2}x  cos ^{2}x]
Solution to Question 2:

Factor the denominator
[sin ^{4}x  cos ^{4}x] / [sin ^{2}x  cos ^{2}x]
= [sin ^{2}x  cos ^{2}x][sin ^{2}x + cos ^{2}x] / [sin ^{2}x  cos ^{2}x]

and simplify
= [sin ^{2}x + cos ^{2}x] = 1
Questions 3:
Simplify the following trigonometric expression.
[sec(x) sin ^{2}x] / [1 + sec(x)]
Solution to Question 3:

Substitute sec (x) that is in the numerator by 1 / cos (x) and simplify.
[sec(x) sin ^{2}x] / [1 + sec(x)]
= sin ^{2}x / [ cos x (1 + sec (x) ]
= sin ^{2}x / [ cos x + 1 ]

Substitute sin ^{2}x by 1  cos ^{2}x , factor and simplify.
= [ 1  cos ^{2}x ] / [ 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 ^{2}x
Questions 5:
Simplify the following trigonometric expression.
sin ^{2}x  cos ^{2}x sin ^{2}x
Solution to Question 5:

Factor sin ^{2}x out, group and simplify
sin ^{2}x  cos ^{2}x sin ^{2}x
= sin ^{2}x ( 1  cos ^{2}x )
= sin ^{4}x
Questions 6:
Simplify the following trigonometric expression.
tan ^{4}x + 2 tan ^{2}x + 1
Solution to Question 6:

Note that the given trigonometric expression can be written as a square
tan ^{4}x + 2 tan ^{2}x + 1
= ( tan ^{2}x + 1) ^{2}

We now use the identity 1 + tan ^{2}x = sec ^{2}x
= ( sec ^{2}x ) ^{2} = sec ^{4}x
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 ^{2}x]
= 2 / sin ^{2}x = 2 csc ^{2}x
Questions 8:
Write sqrt( 4  4 sin ^{2}x ) without square root for Pi / 2 < x < Pi.
Solution to Question 8:

Factor, and substitute 1  sin ^{2}x by cos ^{2}x
sqrt( 4  4 sin ^{2}x )
= sqrt[ 4(1  sin ^{2}x ) ]
= 2 sqrt[ cos ^{2}x ]
= 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 ^{4}x] / [1 + sin ^{2}x]
Solution to Question 9:

Factor the denominator, and simplify
[1  sin ^{4}x] / [1 + sin ^{2}x]
= [1  sin ^{2}x] [1 + sin ^{2}x] / [1 + sin ^{2}x]
= [1  sin ^{2}x] = cos ^{2}x
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 ^{2}x ]
= 2 / cos ^{2}x = 2 sec ^{2}x
Questions 11:
Add and simplify.
cos x  cos x sin ^{2}x
Solution to Question 11:

factor cos x out
cos x  cos x sin ^{2}x
= cos x (1  sin ^{2}x)
= cos x cos ^{2}x = cos ^{3}x
Questions 12:
Simplify the following expression.
tan ^{2}x cos ^{2}x + cot ^{2}x sin ^{2}x
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 ^{2}x cos ^{2}x + cot ^{2}x sin ^{2}x
= (sin x / cos x) ^{2} cos ^{2}x + (cos x / sin x) ^{2} sin ^{2}x

and simplify
= sin ^{2}x + cos ^{2}x = 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 ^{2}x / sin x
= 1 / sin x  cos ^{2}x / sin x
= (1  cos ^{2}x) / sin x
= sin ^{2}x / sin x
= sin x
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