How can Jamie rewrite the expression start fraction 1 over 1 minus sine theta end fraction so that the fraction has cosine squared theta in the denominator?

Question

(1 point)
Responses

She can multiply the numerator and denominator by cos θ.
She can multiply the numerator and denominator by cos θ .

She can multiply the numerator and denominator by tan θ.
She can multiply the numerator and denominator by tan θ .

She can multiply the numerator and denominator by 1 minus cosine theta.
She can multiply the numerator and denominator by Image with alt text: 1 minus cosine theta .

She can multiply the numerator and denominator by 1 + sin θ.
She can multiply the numerator and denominator by 1 + sin θ .

Answer 1

To rewrite the expression \(\frac{1}{1 - \sin \theta}\) so that the denominator contains \(\cos^2 \theta\), we can use the identity:

\[ 1 - \sin^2 \theta = \cos^2 \theta \]

This means we can express \(1 - \sin \theta\) in a form that involves \(\cos^2 \theta\) by multiplying the numerator and denominator by \(1 + \sin \theta\).

Doing this gives:

\[ \frac{1}{1 - \sin \theta} \cdot \frac{1 + \sin \theta}{1 + \sin \theta} = \frac{1 + \sin \theta}{(1 - \sin \theta)(1 + \sin \theta)} = \frac{1 + \sin \theta}{1 - \sin^2 \theta} = \frac{1 + \sin \theta}{\cos^2 \theta} \]

Thus, the correct response is:

She can multiply the numerator and denominator by \(1 + \sin \theta\).