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. Given is 1 plus the cosecant of theta times 1 minus the cosecant of theta. Now here recognizing that I have a 1 plus the cosecant of theta times 1 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 1 minus the cosecant squared of theta. Now I have a squared trick function. And looking at my pythagorean identities the only one of them that has the cosecant squared is this last one. 1 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 1, 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 let on that right side, leaving me on the left side with 1 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 of 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 Identities

Introduction to Trigonometric Identities

Precalculus

12. Trigonometric Identities

Introduction to Trigonometric Identities

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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

Recognize that the expression $(1 + \csc\theta)(1 - \csc\theta)$ is a difference of squares. This can be rewritten using the identity $a^2 - b^2 = (a - b)(a + b)$.

Apply the difference of squares identity: $(1 + \csc\theta)(1 - \csc\theta) = 1^2 - (\csc\theta)^2 = 1 - \csc^2\theta$.

Recall the Pythagorean identity for cosecant: $\csc^2\theta = 1 + \cot^2\theta$.

Substitute the Pythagorean identity into the expression: $1 - \csc^2\theta = 1 - (1 + \cot^2\theta)$.

Simplify the expression: $1 - 1 - \cot^2\theta = -\cot^2\theta$.

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