Section 5.1 Verifying Trigonometric Identities

Overview In Chapter 4, we developed several classes of trigonometric identities: 1.Quotient 2.Reciprocal 3.Pythagorean 4.Even-Odd Identities are true for all values of x for which the trig functions are defined.

The Fundamental Identities

Verification When we verify a trig identity, we show that one side of the identity can be simplified so that it is identical to the other side.

Rules, Guidelines, and Suggestions 1.Start with the side that appears to be more complicated. 2.Re-write trig functions in terms of sines and cosines. 3.Apply fundamental identities. 4.Use algebraic techniques such as factoring or combining like terms. 5.Use arithmetic techniques such as finding a common denominator, separating fractional terms, or multiplying by a conjugate. 6.Do NOT move terms from one side to the other! 7.Frowned on by many: As a last resort, work on both sides separately.

Example 1: Changing to Sines and Cosines to Verify an Identity Verify the identity: Divide the numerator and the denominator by the common factor. Multiply the remaining factors in the numerator and denominator. The identity is verified.

Example 2: Using Factoring to Verify an Identity Verify the identity: Factor sin x from the two terms. Multiply. The identity is verified.

Example 3: Combining Fractional Expressions (with common denominator) to Verify an Identity Verify the identity: The least common denominator is sin x(1 + cos x) Use FOIL to multiply (1 + cos x)(1 + cos x)

Example 3: (continued) Add the numerators. Put this sum over the LCD. Regroup terms in the numerator. Add constant terms in the numerator. Verify the identity:

Example 3: (continued) Verify the identity: Factor and simplify. Factor out the constant term. The identity is verified.

Example 4: Using a Pythagorean Identity to Verify an Identity

Example 5 : Separating a Single-Term quotient into Two Terms to Verify an Identity

Examples—Verify The Following

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