516 CHAPTER 7 Analytic Trigonometry Establishing an Identity Establish the identity: α β α β α β ( ) − = + cos sin sin cot cot 1 Solution EXAMPLE 6 α β α β α β α β α β α β α β α β α β α α β β α β ( ) − = + = + = ⋅ + = + cos sin sin cos cos sin sin sin sin cos cos sin sin sin sin sin sin cos sin cos sin 1 cot cot 1 Difference Formula for cosine Proof α β α β α β α β α β α β α β ( ) ( ) ( ) + = + + = + − tan sin cos sin cos cos sin cos cos sin sin Now divide the numerator and the denominator by α β cos cos. tan sin cos cos sin cos cos cos cos sin sin cos cos sin cos cos cos cos sin cos cos cos cos cos cos sin sin cos cos sin cos sin cos 1 sin cos sin cos tan tan 1 tan tan α β α β α β α β α β α β α β α β α β α β α β α β α β α β α β α α β β α α β β α β α β ( ) + = + − = + − = + − ⋅ = + − ■ Proof Use the Sum Formula for α β ( ) + tan and Even–Odd Properties to get the Difference Formula for the tangent function. α β α β α β α β α β α β ( ) ( ) [ ] ( ) ( ) − = + − = + − − − = − + tan tan tan tan 1 tan tan tan tan 1 tan tan ↑ θ θ ( ) − =− tan tan ■ In Words Formula (6) states that the tangent of the sum of two angles equals the tangent of the first angle plus the tangent of the second angle, all divided by 1 minus their product. Use the identity θ θ θ = tan sin cos and the Sum Formulas for α β ( ) + sin and α β ( ) + cos to derive a formula for α β ( ) + tan . 2 Use Sum and Difference Formulas to Establish Identities Now Work PROBLEMS 49 AND 61 We have proved the following results: Now Work PROBLEM 35(d) THEOREM Sum and Difference Formulas for the Tangent Function i tan tan tan 1 tan tan α β α β α β ( ) + = + − (6) i tan tan tan 1 tan tan α β α β α β ( ) − = − + (7) Establishing an Identity Establish the identity: θ π θ ( ) + = tan tan Solution EXAMPLE 7 θ π θ π θ π θ θ θ ( ) + = + − = + − ⋅ = tan tan tan 1 tan tan tan 0 1 tan 0 tan Example 7 verifies that the tangent function is periodic with period π.
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