Verifying Trigonometric Identities Verifying Trigonometric Identities
This tutorial shows how to verify trigonometric identities and their use in trigonometric expressions. Examples are used to show a deeper understanding of the verification techniques for trigonometric expressions. The verification of trigonometric identities would not be possible simplification techniques were not used. Being able to use algebra reducing skills is important when verifying trigonometric identities. 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: Know More About Antiderivative Of CSC
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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 .
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= [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.
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Thank You
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