18 Verifying Trigonometric Identities

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1 Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan 18 Verifying Trigonometric Identities In this section, you will learn how to use trigonometric identities to simplify trigonometric expressions. Equations such as (x 2)(x + 2) x 2 4 or x2 1 x 1 x + 1 are referred to as identities. An identity is an equation that is true for all values of x for which the expressions in the equation are defined. For example, the equation (x 2)(x + 2) x 2 4 is defined for all real numbers x. The equation x 2 1 x 1 x + 1 is true for all real numbers x 1. We have already seen many trigonometric identities. For the sake of completeness we list these basic identities: Reciprocal Identities sin x 1 1 csc x sec x csc x 1 sec x 1 sin x tan x 1 tan x 1 cot x cot x Quotient identities tan t sin t ; cot t cos t cos t sin t Pythagorean identities cos 2 x + sin 2 x tan 2 x sec 2 x 1 + cot 2 x csc 2 x 1

2 Even-Odd identities sin ( x) sin x cos ( x) csc ( x) csc x sec ( x) sec x tan ( x) tan x cot ( x) cot x Simplifying Trigonometric Expressions Some algebraic expressions can be written in different ways. Rewriting a complicated expression in a much simpler form is known as simplifying the expression. There are no standard steps to take to simplify a trigonometric expression. Simplifying trigonometric expressions is similar to factoring polynomials: by trial and error and by experience, you learn what will work in which situations. To simplify algebraic expressions we used factoring, common denominators, and other formulas. We use the same techniques with trigonometric expressions together with the fundamental trigonometric identities listed above. Example 18.1 Simplify the expression sec2 θ 1. Using the identity 1 + tan 2 θ we find Example 18.2 Simplify the expression: tan2 θ 1 tan2 θ sin2 θ cos 2 θ cos2 θ sin 2 θ sin θ + 1+cos θ. 1+cos θ sin θ Taking common denominator and using the identity cos 2 θ + sin 2 θ 1 we find 2

3 sin θ 1 + cos θ cos θ sin θ (1 + cos θ)2 + sin 2 θ sin θ(1 + cos θ) 2(1 + cos θ) sin θ(1 + cos θ) 2 csc θ Example 18.3 Simplify the expression: (sin x )(sin x + ). Multiplying we find Example 18.4 Simplify + tan x sin x. (sin x )(sin x + ) sin 2 x cos 2 x Using the quotient identity tan x sin x and the Pythagorean identity cos2 x+ sin 2 x 1 we find + tan x sin x + sin x sin x cos2 x + sin 2 x 1 sec x. Establishing Trigonometric Identities A trigonometric identity is a trigonometric equation that is valid for all values of the variable for which the expressions in the equation are defined. How do you show that a trigonometric equation is not an identity? All you need to do is to show that the equation does not hold for just one value of the variable. For example, the equation sin x + 1 3

4 is not an identity since for x π we have 4 sin π 4 + cos π To verify that an equation is an identity, we start by simplifying one side of the equation and end up with the other side. One of the common methods for establishing trigonometric identities is to start with the side containing the more complicated expression and, using appropriate basic identities and algebraic manipulations, such as taking a common denominator, factoring and multiplying by a conjugate, to arrive at the other side of the equality. Example 18.5 Establish the identity: 1+sec θ sec θ sin2 θ 1 cos θ. Using the identity cos 2 θ + sin 2 θ 1 we have sin 2 θ 1 cos θ 1 cos2 θ 1 cos θ (1 cos θ)(1 + cos θ) 1 cos θ Example 18.6 Show that sin θ cos θ is not an identity. 1 + cos θ cos θ(1 + sec θ) 1 + sec θ sec θ Letting θ π 2 we get 1 sin π 2 cos π 2 0. Example 18.7 Verify the identity: (sec x ) sin 2 x. The left-hand side looks more complex then the right-hand side, so we start with it and try to transform it to the right-hand side. 4

5 (sec x ) sec x cos 2 x 1 cos2 x 1 cos 2 x sin 2 x Example 18.8 Verify the identity: 2 tan x sec x 1 1 sin x 1 1+sin x. Starting from the right-hand side to obtain 1 1 sin x sin x (1 + sin x) (1 sin x) (1 sin x)(1 + sin x) 2 sin x 1 sin 2 x Example 18.9 Verify the identity: 1 sin x 2 sin x cos 2 x 2 sin x 1 sec x + tan x. 2 tan x sec x Using the conjugate of 1 sin x to obtain 1 sin x (1 + sin x) (1 sin x)(1 + sin x) + sin x 1 sin 2 x + sin x cos 2 x 1 + sin x sec x + tan x. 5