Graph each expression and use the graph to make a conjecture, ...

Question

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(cos x sin 2x)/1 + cos 2x)

Accepted Answer

Hello, everyone. We are asked to simplify the given expression. And we want to express our answer in terms of X only. We are given the sine of X divided by one minus the cosine of two X. Our answer choices are a, the Kent of X divided by four B. Kosic can of XC sine of X divided by two D kosi can of X divided by two. First, I'm gonna rewrite the expression we're given. So the sign of X divided by one minus the cosine of two X thinking about my identities. The first identity I notice would help me here is that the cosine of two X is equal to the cosine squared of X minus the sine squared of X. So I am going to plug that in to the denominator. So we will have the sine of X divided by one minus. And I'm gonna put it in parenthesis for the time being the cosine squared of X minus the sine squared of X. We need to distribute the negative sign into the parentheses in the denominator. So we will have the sine of X divided by one minus the cosine squared of X plus the sine squared of X. Looking at this, I noticed that if I put parentheses around one minus cosine squared of X, that I have another identity building itself because one minus the cosine squared of X is equal to the sine squared of X. So knowing that I'm going to place sine squared of X in for the parentheses, one minus the cosine squared of X. So this will give me sine X divided by sine squared of X plus sine squared of X. I want to simplify this and I'm actually gonna move it up to the next column. So I will have the sign of X divided by to sign squared of X, I want to reduce this fraction. And since the sine squared of X is the same as saying the sine of X multiplied by the sin of X, I can reduce the numerators of X with one of the sine of XS from the denominator. And this will give me one divided by two sign of X. And if I separate this, I could think of this as one half multiplied by one divided by the sine of X. One divided by the sine of X is the same as the Kosi can of X. So I could say this is one half multiplied by the cosecant of X which can be simplified to KC can of X divided by two. And that is where I would leave this. So our original expression is equal to the KC can of X divided by two which is answer choice D have a nice day.

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(cos x sin 2x)/1 + cos 2x)

Key Concepts

Trigonometric Identities

Graphing Trigonometric Functions

Algebraic Verification

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