Verify Trigonometric Identities

How to verify trigonometric identities? Several examples with detailed solutions are presented. Since we will make use of the basic trigonometric identities, a list of these Trigonometric Identities is available in this site.



No single method works for all identities. However following certain steps might help. To verify an identity, you may start by transforming the more complicated side into the other using basic identities. Or you may transform the two sides into one same expression.

Example 1:
Verify the identity cos x * tan x = sin x

Solution to Example 1:

  • We start with the left side and transform it into sin x. Use the identity tan x = sin x / cos x in the left side.

    cos x * tan x = cos x * (sin x / cos x) = sin x


Example 2:
Verify the identity cot x * sec x * sin x = 1

Solution to Example 2

  • Use the identities cot x = cos x / sin x and sec x = 1/ cos x in the left side.

    cot x * sec x * sin x = (cos x / sin x) * (1/ cos x) * sin x

  • Simplify to obtain.

    (cos x / sin x) * (1/ cos x) * sin x = 1


Example 3:

Verify the identity [ cot x - tan x ] / [sin x * cos] = csc2x - sec2x



Solution to Example 3

  • We use the identities cot x = cos x / sin x and tan x = sin x / cos x to transform the left side as follows.

    [ cot x - tan x ] / [sin x * cos] = [cos x / sin x - sin x / cos x] / [sin x * cos]

  • Rewrite the upper part of the above with a common denominator .

    = [cos 2x / sin x * cos x- sin 2x / cos x * sin x] / [sin x * cos]

    = [cos 2x - sin 2x] / [sin x * cos]2 (expression 1)

  • We now transform the right side using the identities csc x = 1 / sin x and sec x = 1 / cos x.

    csc2x - sec2x = (1/sin x)2 - (1/cos x)<2
  • We now rewrite the above expression with a common denominator = [ cos2x - sin2x ] / [sin x * cos]2 (expression 2)

  • We have transformed the left side to expression 1 and the right side to expression 2. These two expressions are equal. We have verified the given identity.




Exercises
  1. Verify the identity sin x + cos x * cot x = csc x.

  2. Verify the identity [csc x / (1 + csc x) - csc x / (1 - csc x)] = 2*sec2x.



Trigonometric Identities and Their Applications and Trigonometric Formulas and Their Applications


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Updated: 2 April 2013

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