Chapter 5.2

5.2 Verifying Trigonometric Identities In this section you will study techniques for verifying trigonometric identities. In 5.3 you will study techniques for solving trigonometric identities *The key to both is your ability to use the fundamental identities and the rules of algebra to rewrite the trigonometric expressions. Remember: Solving an equation is finding the values that make a statement true. For example: is true for This is an example of a conditional equation.

5.2 Verifying Trigonometric Identities An equation that is true for all real values in the domain of the variable is an identity. For example: is true for all real numbers x. So, it is an identity. There is no well-defined set of rules to follow in verifying trigonometric identities, but there are some guidelines. Work with one side of the equation at a time. It is often better to work with the more complicated side first. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. (In other words, look for ways to get the expression into something you recognize)

5.2 Verifying Trigonometric Identities 3. Look for opportunities to use the fundamental identities. Remember: Sine and cosine pair up well, as do secants and tangents, and cosecants and cotangents. 4. If step 1 through 3 are not helping try converting all terms to sines and cosines. 5. MAKE AN ATTEMPT! Always try something, it might provide insight for you. Note: Since you are trying to verify one side is the same as the other you can not cross multiply or add the same quantity to both sides. Do not assume both sides are equal, that is what you are trying to verify! O.K. Let’s get started!

5.2 Verifying Trigonometric Identities Verify the identity Start with the left side because it is more complicated Pythagorean identity Simplify *Remember* There can be more than one way to verify an identity Reciprocal identity Quotient identity Simplify

5.2 Verifying Trigonometric Identities Try #5 pg. 3531 Verify the identity

5.2 Verifying Trigonometric Identities Combining Fractions Before Using Identities Verify the identity Add fractions Simplify Pythagorean identity Reciprocal identity

5.2 Verifying Trigonometric Identities Try #31 pg. 3542 Verify the identity algebraically

5.2 Verifying Trigonometric Identities Verify the identity Apply the identities before you multiply Reciprocal identity Rule of exponents Quotient identity

5.2 Verifying Trigonometric Identities Try #41 pg. 3543 Verify the identity algebraically

5.2 Verifying Trigonometric Identities Verify the identity Multiply numerator and denominator by (1+sin x) Multiply Pythagorean identity Separate fractions Simplify Identities

5.2 Verifying Trigonometric Identities Try #47 pg. 3544 Verify the identity algebraically

5.2 Verifying Trigonometric Identities Work with each side separately Verify the identity Left Side Right Side

5.2 Verifying Trigonometric Identities Try #49 pg. 3545 Verify the identity algebraically

5.2 Verifying Trigonometric Identities Enriched Pre-Calculus Verify each identity Rewrite as separate factors Pythagorean identity Multiply Rewrite as separate factors Pythagorean identity Multiply

5.2 Verifying Trigonometric Identities Try #63 pg. 3556 Verify the identity

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