Use the Pythagorean identities to rewrite the expression as a single term.(1+cscθ)(1−cscθ)\left(1+\csc\theta\right)\left(1-\csc\theta\right)(1+cscθ)(1−cscθ)

− cot ⁡ 2 θ -\cot^2\theta − cot 2 θ

Use the Pythagorean identities to rewrite the expression as a single term. ( 1 + csc θ ) ( 1 − csc θ ) \left(1+\csc\theta\right)\left(1-\csc\theta\right) ( 1 + csc θ ) ( 1 − csc θ )

In this problem we're asked to use the Pythagorean identities to rewrite the expression as a single term. So let's go ahead and get started. The expression that we're given is one plus the cosecant of theta times one minus the cosecant of theta. Now here recognizing that I have a one plus the cosecant of theta times one minus that same function the cosecant of theta, I can use my difference of squares to recognize that this multiplied together will end up being one minus the cosecant squared of theta. Now I have a a squared trig function and looking at my pythagorean identities the only one of them that has the cosecant squared is this last one. One plus the cotangent squared of theta is equal to the cosecant squared of theta, but this isn't in the exact form that I need it to be in, so let's play around with this a little bit. If I subtract the cotangent squared of theta from both sides it will cancel on that left side leaving me with just one and that's equal to the cosecant squared of theta minus the cotangent squared of theta. But this still isn't in the exact form that I want it to be in. So what else can I do here? Well, if I subtract the cosecant squared of theta from both sides, it will cancel on that left on that right side, leaving me on the left side with one minus the cosecant squared of theta. Then all that I'm left with on the right side is the negative cotangent squared of theta. So here, that's exactly the expression that I'm looking for, the one minus the cosecant squared theta. So this one minus the cosecant squared of theta is simply equal to negative the cotangent squared of theta. Now as you continue to work through these problems it's going to become more and more obvious what these look like in different forms. So while right now you might have to go through this whole process of rewriting and manipulating your Pythagorean identities, soon you'll be able to recognize it right away. So here one minus the cosecant squared of theta ends up just being negative cotangent squared of theta and that's my final answer. Let me know if you have any questions. Thanks for watching and I'll see you in the next one.

Use the Pythagorean identities to rewrite the expression as a single term. ( 1 + csc ⁡ θ ) ( 1 − csc ⁡ θ ) \left(1+\csc\theta\right)\left(1-\csc\theta\right) ( 1 + csc θ ) ( 1 − csc θ )

− cot ⁡ 2 θ -\cot^2\theta − cot 2 θ

− csc ⁡ 2 θ -\csc^2\theta − csc 2 θ

cot ⁡ 2 θ \cot^2\theta cot 2 θ

12. Trigonometric Functions

Trigonometric Identities

Business Calculus

12. Trigonometric Functions

Trigonometric Identities

Struggling with Business Calculus?

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Multiple Choice

Use the Pythagorean identities to rewrite the expression as a single term.
$\left(1+\csc\theta\right)\left(1-\csc\theta\right)$

$-\csc^2\theta$

$\cot^2\theta$

$-\cot^2\theta$

Verified step by step guidance

Step 1: Recognize that the given expression involves trigonometric identities, specifically the Pythagorean identities. Recall that one of the Pythagorean identities is: