Expert Answer to "How to simplify cot⁡(θ)csc⁡(θ)−sin⁡(θ)?"

Jace Nelson

Answered question

2023-01-08

How to simplify

\frac{\cot (θ)}{\csc (θ) - \sin (θ)}

brenosa0p7

Beginner2023-01-09Added 6 answers

= \frac{\frac{\cos θ}{\sin θ}}{\frac{1}{\sin θ} - \sin θ}

= \frac{\frac{\cos θ}{\sin θ}}{\frac{1}{\sin θ} - \frac{\sin^{2} θ}{\sin θ}}

= \frac{\frac{\cos θ}{\sin θ}}{\frac{1 - \sin^{2} θ}{\sin θ}}

= \frac{\frac{\cos θ}{\sin θ}}{\frac{\cos^{2} θ}{\sin θ}}

= \frac{\cos θ}{\sin θ} \times \frac{\sin θ}{\cos^{2} θ}

= \frac{1}{\cos θ}

= \sec θ

Jillian Vang

Beginner2023-01-10Added 1 answers

Since

\cot θ = \frac{\cos θ}{\sin θ} and \csc θ = \frac{1}{\sin θ}

, the phraseexpression changes to:

\frac{\frac{\cos θ}{\sin θ}}{\frac{1}{\sin θ} - \sin θ}

that's

\frac{\frac{\cos θ}{\sin θ}}{\frac{1 - \sin^{2} θ}{\sin θ}}

;
then, since

1 - \sin^{2} θ = \cos^{2} θ

, the phraseexpression changes to:

Undefined control sequence \cancel

= \frac{1}{\cos θ} = \sec θ

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