691 7.2 Verifying Trigonometric Identities EXAMPLE 3 Verifying an Identity (Working with One Side) Verify that the following equation is an identity. tan t - cot t sin t cos t = sec2 t - csc2 t SOLUTION We transform the more complicated left side to match the right side. tan t - cot t sin t cos t Left side of given equation = tan t sin t cos t - cot t sin t cos t a - b c = a c - b c = tan t # 1 sin t cos t - cot t # 1 sin t cos t a b = a # 1 b = sin t cos t # 1 sin t cos t - cos t sin t # 1 sin t cos t tan t = sin t cos t ; cot t = cos t sin t (This is Hint 3.) = 1 cos2 t - 1 sin2 t Multiply. = sec2 t - csc2 t 1 cos2 t = sec2 t; 1 sin2 t = csc2 t S Now Try Exercise 53. EXAMPLE 4 Verifying an Identity (Working with One Side) Verify that the following equation is an identity. cos x 1 - sin x = 1 + sin x cos x SOLUTION We work on the right side, using Hint 6 in the list given earlier. 1 + sin x cos x Right side of given equation = 11 + sin x211 - sin x2 cos x11 - sin x2 Multiply by 1 in the form 1 - sin x 1 - sin x . (This is Hint 6.) = 1 - sin2 x cos x11 - sin x2 1x + y21x - y2 = x2 - y2 = cos2 x cos x11 - sin x2 1 - sin2 x = cos2 x = cos x # cos x cos x11 - sin x2 a2 = a # a = cos x 1 - sin x Divide out the common factor to write in lowest terms. S Now Try Exercise 59. Verifying Identities by Working with Both Sides If both sides of an identity appear to be equally complex, the identity can be verified by working independently on the left side and on the right side, until each side is changed into some common third result. Each step, on each side, must be reversible. With all steps reversible, the procedure is as shown in the margin. left = right common third expression
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