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6.1A Reciprocal, Quotient, and Pythagorean Identities Recall the basic definitions for a unit circle: y sin r x cos r y tan x sin . Prove that tan cos y x y x r r y r r x y x L.S. = R.S. A trigonometric identity is a trigonometric equation that is true for all permissible values of the variable in the expressions on both sides of the equation. Both sides of the equation have the same value for all permissible values of the Math 30-1 1 variable. Trigonometric Identities Reciprocal Identities 1 csc sin sec 1 sin csc 1 cos cot 1 cos sec 1 tan 1 tan cot Quotient Identities sin tan cos Pythagorean Identities cos cot sin sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2 sin2 = 1 - cos2 cos2 = 1 - sin2 tan2 = sec2 - 1 cot2 = csc2 - 1 Math 30-1 2 Simplifying Trigonometric Expressions Identities can be used to simplify trigonometric expressions. Simplify to a single trigonometric expression. a) b) sin cot cos sin sin cos cot 2 1 sin2 cos 2 2 sin cos 2 cos 2 1 2 sin cos2 1 2 sin csc 2 Math 30-1 3 Simplify to a single trigonometric expression. c) d) sin 2 cos 2 tan 2 1 tan 2 1 cos 2 sin cos sin 2 sin cos sec 2 sin cos tan Math 30-1 4 Simplifing Trigonometric Expressions e) (1 + tan x)2 - 2 sin x sec x 1 cos x sin x 2 1 2 tan x tan x 2 cos x 2 (1 tan x) 2 sin x 1 tan2 x 2tanx 2 tanx sec2 x Math 30-1 csc x f) tan x cot x 1 sin x sin x cos x cos x sin x 1 sin x sin 2 x cos 2 x sin xcos x 1 sin x 1 sin x cos x 1 sin x cos x sin x 1 cos x 5 Proving an Identity using a Two Column Proof 1 1 2 2 cs c x 1 cos x 1 cos x (1 cos x) (1 cos x) 2 csc2 x (1 cos x)(1 cos x) 2 (1 cos 2 x) 2 sin 2 x 2 csc 2 x L.S. = R.S. Math 30-1 6 Possible Strategies for Simplifying using Identities Trigonometric Simplifications: Look for familiar trig relationships and substitute Rewrite in terms of sine or cosine. If the expression contains squared terms, try using the Pythagorean Identities. Algebraic Simplifications: Multiply, expand, factor, reduce or square. Common denominator to add or subtract. Multiply by the conjugate of a binomial. 7