689 7.2 Verifying Trigonometric Identities LOOKING AHEAD TO CALCULUS Trigonometric identities are used in calculus to simplify trigonometric expressions, determine derivatives of trigonometric functions, and change the form of some integrals. Hints for Verifying Identities 1. Learn the fundamental identities. Whenever one side of a fundamental identity is given, the other side should come to mind. Also, be aware of equivalent forms of the fundamental identities. For example, sin2 u = 1 - cos2 u is an alternative form of sin2 u + cos2 u = 1. 2. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side. 3. It is sometimes helpful to express all trigonometric functions in the equation in terms of sine and cosine and then simplify the result. 4. Usually, any factoring or indicated algebraic operations should be performed. These algebraic identities are often used in verifying trigonometric identities. x2 + 2xy + y2 = 1x + y22 x2 - 2xy + y2 = 1x - y22 x3 - y3 = 1x - y21x2 + xy + y22 x3 + y3 = 1x + y21x2 - xy + y22 x2 - y2 = 1x + y21x - y2 For example, the expression sin2 x + 2 sin x + 1 can be factored as 1sin x + 122. The sum or difference of two trigonometric expressions can be found in the same way as any other rational expression. For example, 1 sin u + 1 cos u = 1 # cos u sin u cos u + 1 # sin u cos u sin u Write with the LCD. = cos u + sin u sin u cos u . a c + b c = a + b c 5. When selecting substitutions, keep in mind the side that is not changing, because it represents the goal. For example, to verify that the equation tan2 x + 1 = 1 cos2 x is an identity, think of an identity that relates tan x to cos x. In this case, because sec x = 1 cos x and sec2 x = tan2 x + 1, the secant function is the best link between the two sides. 6. If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 - sin x would give 1 - sin2 x, which could be replaced with cos2 x. Similar procedures apply for 1 - sin x, 1 + cos x, and 1 - cos x. Verifying Identities by Working with One Side Avoid the temptation to use algebraic properties of equations to verify identities. One strategy is to work with one side and rewrite it to match the other side.
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