Simplify Trigonometric Expressions
Questions With Answers

Use trigonometric identities and formulas to simplify trigonometric expressions. The trigonometric identities and formulas in this site might be helpful to solve the questions below.

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