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Section 8.3 Trigonometric Identities WHAT IS AN IDENTITY? Two functions f and g are said to be identically equal if f (x) = g(x) for every value of x for which both functions are defined. Such an equation is referred to as an identity. An equation that is not an identity is called a conditional equation. EXAMPLES Simplify the following. 1. (cot θ)(sec θ) by rewriting in terms of sine and cosine sin 1 cos 2. by multiplyin g by 1 cos 1 cos 1 1 by getting a common denominator 3. 1 cos 1 cos 2 cos 1 4. Factor and simplify 2 cos cos TIPS FOR VERIFYING A TRIGONOMETRIC IDENTITY 1. Work on the most complex side of the expression first. 2. Perform any indicated operations; i.e., add fractions, squaring, etc. 3. Factoring may also help. 4. Rewrite one side in terms of sine and cosine only. 5. Multiply the numerator and denominator of a fraction by the same factor. 6. Work only on one side at a time. EXAMPLES Verify the following trigonometric identities. 1. cot x 1 csc x (cos x sin x) csc x cos x 2. sec x sin x cot x 1 2 2 3. tan x (1 cot x) 2 1 sin x cos x 1 sin x 4. 1 sin x cos x