SECTION 7.4 Trigonometric Identities 505 STUDY TIP In the solution for (b), a “well-chosen 1” is used to obtain a useful form of the Pythagorean Identity in the denominator. Although it is not always obvious what form of 1 to use, through practice you will learn the best forms to try. j NOTE A graphing utility cannot be used to establish an identity—identities must be established algebraically. A graphing utility can be used to provide evidence of an identity. For example, if we graph Y csc tan 1 θ θ = ⋅ and Y sec , 2 θ = the graphs appear to be the same, providing evidence that Y Y . 1 2 = j Using Algebraic Techniques to Simplify Trigonometric Expressions (a) Simplify cot csc θ θ by rewriting each trigonometric function in terms of sine and cosine functions. (b) Show that cos 1 sin 1 sin cos θ θ θ θ + = − by multiplying the numerator and denominator by 1 sin .θ − (c) Simplify u u u u u 1 sin sin cot cos cos + + − by rewriting the expression as a single ratio. (d) Simplify v v v v sin 1 tan sin tan 2 − − by factoring. Solution EXAMPLE 1 (a) cot csc cos sin 1 sin cos sin sin 1 cos θ θ θ θ θ θ θ θ θ = = ⋅ = (b) cos 1 sin cos 1 sin 1 sin 1 sin cos 1 sin 1 sin cos 1 sin cos 1 sin cos 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) + = + ⋅ − − = − − = − = − (c) + + − = + ⋅ + − ⋅ = + + − = + = + ⋅ = + = = u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u 1 sin sin cot cos cos 1 sin sin cos cos cot cos cos sin sin cos sin cos cot sin cos sin sin cos cos cot sin sin cos cos cos sin sin sin cos cos cos sin cos 2cos sin cos 2 sin (d) v v v v v v v v v v sin 1 tan sin tan sin 1 sin 1 tan sin 1 sin 1 tan 2 ( )( ) ( ) − − = + − − = + æ Multiply by a well-chosen θ θ − − 1: 1 sin 1 sin . æ =u u u cot cos sin Now Work PROBLEMS 11, 13, AND 15 2 Establish Identities In the examples that follow, the directions read “Establish the identity . . . .” This is accomplished by starting with one side of the given equation (usually the side containing the more complicated expression) and, using appropriate basic identities and algebraic manipulations, arriving at the other side. The selection of appropriate basic identities to obtain the desired result is learned only through experience and lots of practice.
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