__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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