685 7.1 Fundamental Identities The simplified form of a trigonometric expression contains a minimum number of terms and no quotients, if possible. When simplifying expressions containing the functions tan u, cot u, sec u, or csc u, it is often beneficial to express them in terms of sin u or cos u (or both) and then simplify. EXAMPLE 3 Rewriting an Expression inTerms of Sine and Cosine Write 1 + cot 2 u 1 - csc2 u in terms of sin u and cos u, and then simplify the expression so that no quotients appear. SOLUTION 1 + cot2 u 1 - csc2 u Given expression = 1 + cos2 u sin2 u 1 - 1 sin2 u Quotient identities = a1 + cos2 u sin2 ubsin2 u a1 - 1 sin2 ubsin2 u Simplify the complex fraction by multiplying both numerator and denominator by the LCD. = sin2 u + cos2 u sin2 u - 1 Distributive property: 1b + c2a = ba + ca = 1 -cos2 u Pythagorean identities = - sec2 u Reciprocal identity S Now Try Exercise 59. y2 = −sec 2 x −4 4 11p 4 1 + cot2 x 1 − csc2 x y1 = 11p 4 − CAUTION When working with trigonometric expressions and identities, be sure to write the argument of the function. For example, we would write sin2 u + cos2 u = 1, not sin2 + cos2 = 1. 7.1 Exercises CONCEPT PREVIEW For each expression in Column I, choose the expression from Column II that completes an identity. I 1. cos x sin x = 2. tan x = 3. cos1-x2 = 4. tan2 x + 1 = 5. 1 = II A. sin2 x + cos2 x B. cot x C. sec2 x D. sin x cos x E. cos x The graph supports the result in Example 3. The graphs of y1 and y2 coincide.
RkJQdWJsaXNoZXIy NjM5ODQ=