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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


More trigonometry questions with solutions and answersin this site.


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Updated: 2 April 2013

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