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.

csc x - cot x

Accepted Answer

Hello, everyone. We are asked to simplify the given expression. We want to express our answer in terms of X only. We are given one divided by the KC can of X plus the cotangent of X. Our answer choices are a, the tangent of X divided by two B, the cotangent of X C, the tangent of XD, the cotangent of X divided by two. First, I'm gonna rewrite our given expression. So one divided by the cos of X plus the cotangent of X. I am going to rewrite this one more time in terms of cosign and sign. So I'm going to have one divided by and then Kosi can of X is one divided by the sine of X. And I'm going to add that to the cotangent of X which can be rewritten as the cosine of X divided by the sine of X. So now simplifying my denominator since they have common denominators, my numerator is one. And then in the denominator, I will have one plus the cosine of X divided by the sine of X. I want to rewrite this as a simpler fraction. So one divided by one plus the cosine of X divided by the sine of X. If I multiply by the reciprocal is one multiplied by the sine of X divided by one plus the cosine of X. And this will give us the sine of X divided by one plus the cosine of X. And I recognize that as one of our identities, this is the tangent half angle identity. So that means this can be rewritten as the tangent of X divided by two. And I don't have to do any other rearranging. It just matches in identity we already have. And tangent of X divided by two is answer choice A 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.
csc x - cot x

Key Concepts

Cosecant and Cotangent Functions

Graphing Trigonometric Functions

Verifying Trigonometric Identities

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