Showing that two sides of a potential trigonometric identity are equal for a given value is not enough proof that it is true for all permissible values of the variable. Multiply, expand, factor, reduce or square. Common denominator to add or subtract. Substitute for familiar trig relationships If the expression contains squared terms or 1, -1, try using a Pythagorean Identity. Express in terms of sine or cosine. Multiply by the conjugate of a binomial. Math 30-11

Proving an Equation is an Identity Consider the equation b) Verify that this statement is true for x = 2.4 rad. a) Use a graph to verify that the equation is an identity. c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. a) Math 30-12

b) Verify that this statement is true for x = 2.4 rad. Proving an Equation is an Identity [Cont’d] = Therefore, the equation is true for x = 2.4 rad. L.S. = R.S. Math 30-13

Proving an Equation is an Identity c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. L.S. = R.S. Note the left side of the equation has the restriction sin 2 A - sin A ≠ 0 The right side of the equation has the restriction sin A ≠ 0, or A ≠ 0. Therefore, A ≠ 0,  + 2  n, where n is any integer. sin A(sin A - 1) ≠ 0 sin A ≠ 0 or sin A ≠ 1 Math 30-14

L.S. = R.S. Proving an Identity common denominator Math 30-15

sin 4 x - cos 4 x = 1 - 2cos 2 x = (sin 2 x - cos 2 x)(sin 2 x + cos 2 x) = (sin 2 x - cos 2 x)(1) L.S. = R.S cos 2 x Proving an Identity factoring = 1 - cos 2 x- cos 2 x = 1 - 2cos 2 x Math 30-16

Proving an Equation is an Identity Substitution L.S. = R.S. Math 30-17

Proving an Equation is an Identity Multiplying by Conjugates Math 30-18

Page 314 1a,d, 2a,c, 3c, 4,5, 7, 8, 9, 10, 11a,c, 15, C1, C2 Math 30-19