SECTION 7.4 Trigonometric Identities 507 When sums or differences of quotients appear, it is usually best to rewrite them as a single quotient, especially if the other side of the identity consists of only one term. Establishing an Identity Establish the identity: sin 1 cos 1 cos sin 2csc θ θ θ θ θ + + + = EXAMPLE 6 Solution The left side is more complicated. Start with it and add. sin 1 cos 1 cos sin sin 1 cos 1 cos sin sin 1 2cos cos 1 cos sin sin cos 1 2cos 1 cos sin 2 2cos 1 cos sin 2 1 cos 1 cos sin 2 sin 2csc 2 2 2 2 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + + = + + + ⋅ = + + + + ⋅ = + + + + ⋅ = + + ⋅ = + + ⋅ = = Add the quotients. Multiply out in the numerator. Regroup. θ θ + = sin cos 1 2 2 Factor. Divide out θ +1 cos . Reciprocal Identity Now Work PROBLEM 51 Sometimes it helps to write one side in terms of sine and cosine functions only. Establishing an Identity Establish the identity: v v v v tan cot sec csc 1 + = EXAMPLE 7 Solution v v v v v v v v v v v v v v v v v v v v tan cot sec csc sin cos cos sin 1 cos 1 sin sin cos cos sin 1 cos sin 1 cos sin cos sin 1 1 2 2 + = + ⋅ = + = ⋅ = ↑ Change to sines and cosines. ↑ Add the quotients in the numerator. ↑ Divide the quotients; + = v v sin cos 1. 2 2 Now Work PROBLEM 71 Sometimes, multiplying the numerator and the denominator by an appropriate factor simplifies an expression.
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