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.

cos θ (cos 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 cotangent of X multiplied by the parentheses. Sine X minus C can X our answer choices are a in parentheses, sine X cosine X plus one close parentheses divided by cosine of X B in parentheses, sine X cosine of X minus one, close parentheses divided by the sine of X C in parentheses, sine X cosine X minus one closed parentheses divided by sine squared X D in parentheses, sine X cosine X minus one closed parentheses divided by the cosine of X. So first, I'm going to rewrite my original expression and I have the cotangent of X multiplied by the sine of X minus the C of X. So we want all of this in terms of sign and cosine. So I recall that the cotangent of X is the same as saying the cosine of X divided by the sine of X. And then in the parentheses, the sign of X will remain the sign of X and the sin of X. Our reciprocal identity tells me that that is one divided by the cosine of X. So now let's distribute this into the parentheses and see what we get. So I'm gonna distribute the cosine of X divided by the sine of X multiplied by the sine of X. And this will give me the cosine of X sine of X divided by the sign of X. And then the next term, the cosine of X divided by the sine of X multiplied by negative one, divided by the cosine of X will give me negative cosine of X divided by sine of X cosine of X. All right. So now that we've distributed this, I'm gonna reduce what I can within these fractions. So looking at the first fraction cosine X sine X divided by the sine of X, I can reduce the sine of X with the sine of X. So that leaves my first fraction as no longer a fraction but just the cosine of X in my second fraction, the cosine of X divided by sin of X cosine of X. I can reduce the cosine of X in the numerator with the cosine of X in the denominator. I don't want to leave my numerator completely empty. So I put a placeholder one and my denominator is now the sign of X. So I have the cosine of X minus one divided by the sine of X. I want these to have a common denominator that common denominator is going to be the sign of X. So I'm going to treat the cosine of X as if it's divided by one and multiply both the numerator and denominator by the sine of X. So we will have the sine of X cosine of X divided by the sine of X minus one divided by the sine of X. Now that they both have the same denominator, the sign of X, we can rewrite them as one fraction. So our num readers will be combined as the sin of X cosine of X minus one. And our common denominator was the sign of X. So this is our solution, the sin of X cosine of X minus one in the numerator divided by the sine of X in the denominator. And this matches answer choice B 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.
cos θ (cos θ - sec θ)

Key Concepts

Trigonometric Functions and Their Reciprocal Identities

Expression Simplification Without Quotients

Algebraic Manipulation of Trigonometric Expressions

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