Write each expression in terms of sine and cosine, and then si...

Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sin θ - cos θ) (csc θ + sec θ)

Accepted Answer

Hello, everyone. We are asked to write the following trigonometric expression. In terms of sine and cosine. We want to express our answer in terms of X only. We are given the parentheses, cosine X plus sin X multiplied by the parentheses. C can't X minus KX. Our answer choices are a in the numerator. The parentheses, sine squared X minus cosine squared X divided by sine X B. The numerator in parentheses, sine squared X plus cosine squared X divided by the cosine of X. See, the numerator has a parenthesis of sine squared X minus cosine squared X divided by sine X cosine X D numerator in parentheses of sine squared X plus cosine squared X, all of which is divided by sine X cosine X. All right. I'm gonna rewrite my original expression. So I have the cosine of X plus the sine of X in parentheses multiplied by the C can of X minus the co C can of X. So our first set of parentheses, the cosine of X plus the sine of X is already in terms of sine and cosine. So we'll leave that alone for now. I want to rewrite my second set of parentheses in terms of sign and cosign. So C can't using the reciprocal identity is the same as one divided by the cosine of X and minus koi can of X which again using a reciprocal identity is one divided by the sign of X. So now I have two binomials multiplied together and I'm gonna have to use foil. So if I foil this first will be the cosine of X multiplied by one divided by the cosine of X which will give me the cosine of X divided by the cosine of X outer will give me the cosine of X multiplied by negative one divided by the sine of X. To be negative cosine of X divided by sine of X inner. I will have the sine of X multiplied by one divided by the cosine of X which will give me a positive sine X divided by cosine X. And last I get the sine of X multiplied by negative one divided by the sine of X to give me negative sine of X divided by the sine of X. So let's see what we can simplify. First, I recognize that any value divided by itself equals one. So the cosine of X divided by the cosine of X is one minus the cosine of X divided by the sine of X plus the sine of X divided by the cosine of X and then minus side effects divided by side effects would be one. So combining my like terms, I have a positive one and a negative one. So those will cancel out and I will be left with and I'm gonna rewrite this. So the negatives not in front, I'm left with the sine of X divided by the cosine of X minus the cosine of X divided by the sine of X. So having two fractions, I need to have a common denominator. The common denominator here will be the sine of X multiplied by the cosine of X. So in my first fraction, I'm gonna multiply both the numerator and the denominator by the sine of X. And in the second fraction, I'll be multiplying the numerator and the denominator by the cosine of X. Let me make that a little less messy. And when I do so, my first fraction will become sine squared X divided by sine X cosine X. My second fraction will have a minus sine in between will have the cosine squared X divided by the sine of X cosine of X. Combining those into one fraction. I get the sine squared of X minus the cosine squared of X divided by the sine of X cosine of X. And there's nothing else here. I can simplify that will leave it in terms of sign and cosign so this is our answer sine squared X minus cosine squared X divided by the sine of X cosine of X. And that's intero C have a nice Day.

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sin θ - cos θ) (csc θ + sec θ)

Key Concepts

Trigonometric Functions and Their Reciprocal Identities

Algebraic Manipulation of Trigonometric Expressions

Avoiding Quotients in Trigonometric Simplification

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