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Simplifying Trigonometric Expressions

Convert to sine/cosine and use basic trig identities to simplify.

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Practice Simplifying Trigonometric Expressions
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Simplifying Trigonometric Expressions

How could you write the trigonometric function more simply?


Now that you are more familiar with trig identities, we can use them to simplify expressions. Remember, that you can use any of the identities in the Introduction to Trig Identities concept. Here is a list of the identities again:

Reciprocal Identities: and

Tangent and Cotangent Identities: and

Pythagorean Identities: and

Cofunction Identities: and

Negative Angle Identities: and

Example A

Simplify .

Solution: When simplifying trigonometric expressions, one approach is to change everything into sine or cosine. First, we can change secant to cosine using the Reciprocal Identity.

Now, combine the denominator into one fraction by multiplying 1 by .

Change this problem into a division problem and simplify.

Example B

Simplify .

Solution: With this problem, we need to factor the numerator and denominator and see if anything cancels. The rules of factoring a quadratic and the special quadratic formulas can be used in this scenario.

In the last step, we simplified to the left hand side of the Pythagorean Identity. Therefore, this expression simplifies to 1.

Example C

Simplify .

Solution: First, pull out the GCF.

Now, from the Pythagorean Identities, so simplify further.

Concept Problem Revisit

Notice that the terms in the expression have a common factor of , so start by factoring this common term out.

Now, use the trigonometric identity , substitute, and simplify.

Guided Practice

Simplify the following trigonometric expressions.





1. Use the Cotangent Identity and the Cofunction Identity .

2. Use the Negative Angle Identity and the Tangent Identity.

3. In this problem, you will use several identities.

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Simplify the following expressions.

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