Example 1 Verify a Trigonometric Identity The left-hand side of this identity is more complicated, so transform that expression into the one on the right. Verify that. Pythagorean Identity Reciprocal Identity Simplify.

Example 1 Verify a Trigonometric Identity Answer:

Example 2 Verify a Trigonometric Identity by Combining Fractions The right-hand side of the identity is more complicated, so start there, rewriting each fraction using the common denominator 1 – cos 2 x. Verify that. Distributive Property Common denominator Start with the right hand side of the identity.

Example 2 Verify a Trigonometric Identity by Combining Fractions Simplify. Divide out the common factor of sin x. Simplify. Quotient Identity

Example 2 Verify a Trigonometric Identity by Combining Fractions Answer:

Example 3 Verify a Trigonometric Identity by Multiplying Because the left-hand side of this identity involves a fraction, it is slightly more complicated than the right side. So, start with the left side. Verify that. Multiply the numerator and denominator by the conjugate of sec x – 1, which is sec x + 1. Multiply.

Example 3 Verify a Trigonometric Identity by Multiplying Pythagorean Identity Divide out the common factor of sin x. Multiply by the reciprocal of the denominator. Quotient Identity

Example 3 Verify a Trigonometric Identity by Multiplying Divide out the common factor of cos x. Rewrite the fraction as the sum of two fractions; Reciprocal Identity. Distributive Property

Example 3 Verify a Trigonometric Identity by Multiplying Quotient Identity

Example 3 Verify a Trigonometric Identity by Multiplying Answer:

Example 4 Verify a Trigonometric Identity by Factoring Verify that cos x sec 2 x tan x – cos x tan 3 x = sin x. cos x sec 2 x tan x – cos x tan 3 x = cos x tan x (sec 2 x – tan 2 x) Start with the left- hand side of the identity. Factor. Pythagorean Identity = cos x tan x (1) Quotient Identity = Divide out the common factor of cos x. = sin x

Example 4 Verify a Trigonometric Identity by Factoring Answer:cos x sec 2 x tan x – cos x tan 3 x = cos x tan x (sec 2 x – tan 2 x) = cos x tan x (1) = = sin x

Example 5 Verify an Identity by Working Each Side Separately Both sides look complicated, but there is a clear first step for the expression on the left. So, start with the expression on the left. Verify that. Simplify and apply a Reciprocal Identity. Write as the sum of two fractions.

Example 5 Verify an Identity by Working Each Side Separately Simplify. Factor. Pythagorean Identity From here, it is unclear how to transform 1 + cot x into, so start with the right side and work to transform it into the intermediate form 1 + cot x.

Example 5 Verify an Identity by Working Each Side Separately Divide out the common factor of 1 – cot x. To complete the proof, work backward to connect the two parts of the proof. Simplify and apply a Reciprocal Identity. Multiply by. Write as the sum of two fractions.

Example 5 Verify an Identity by Working Each Side Separately Simplify. Simplify. Pythagorean Identity Answer:

Key Concept 6

Example 6 The equation appears to be an identity because the graphs of the related functions over [–2π, 2π] scl: π by [–1, 3] scl: 1 coincide. Verify this algebraically. Determine Whether an Equation is an Identity A. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.

Example 6 Determine Whether an Equation is an Identity Pythagorean Identity Divide out the common factor of sec x.

Example 6 Answer: Determine Whether an Equation is an Identity Reciprocal Identities Simplify. Quotient Identity

Example 6 Determine Whether an Equation is an Identity B. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.

Example 6 Determine Whether an Equation is an Identity The graphs of the related functions do not coincide for all values of x for which the both functions are defined. When, Y 1 ≈ 1.43 but Y 2 ≈ –0.5. The equation is not an identity.

Example 6 Determine Whether an Equation is an Identity Answer: When, Y 1 ≈ 1.43 but Y 2 = –0.5. The equation is not an identity.