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:

• 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