692 CHAPTER 7 Trigonometric Identities and Equations EXAMPLE 5 Verifying an Identity (Working with Both Sides) Verify that the following equation is an identity. sec a + tan a sec a - tan a = 1 + 2 sin a + sin2 a cos2 a SOLUTION Both sides appear equally complex, so we verify the identity by changing each side into a common third expression. We work first on the left. sec a + tan a sec a - tan a Left side of given equation = 1sec a + tan a2 cos a 1sec a - tan a2 cos a Multiply by 1 in the form cos a cos a . = sec a cos a + tan a cos a sec a cos a - tan a cos a Distributive property = 1 + tan a cos a 1 - tan a cos a sec a cos a = 1 = 1 + sin a cos a # cos a 1 - sin a cos a # cos a tan a = sin a cos a = 1 + sin a 1 - sin a Simplify. On the right side of the original equation, we begin by factoring. 1 + 2 sin a + sin2 a cos2 a Right side of given equation = 11 + sin a22 cos2 a Factor the numerator; x2 + 2xy + y2 = 1x + y22 = 11 + sin a22 1 - sin2 a cos2 a = 1 - sin2 a = 11 + sin a22 11 + sin a211 - sin a2 Factor the denominator; x2 - y2 = 1x + y21x - y2 = 1 + sin a 1 - sin a Write in lowest terms. Thus, sec a + tan a sec a - tan a = 1 + sin a 1 - sin a = 1 + 2 sin a + sin2 a cos2 a , and we have verified that the given equation is an identity. S Now Try Exercise 75. $1111%1111& Left side of given equation $11%11& Common third expression $1111111%1111111& Right side of given equation CAUTION Use the method of Example 5 only if the steps are reversible.
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