Question

Verify the identity. (cos x - sin x + 1) / (cos x + sin x - 1) = (cos x + 1) / sin x To simplify the left side of the identity, multiply the numerator and denominator of the left side of the equation by the same factor. Choose the multiplication below that will be most useful in simplifying the left side of the equation. A. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x + sin x - 1) / (cos x + sin x - 1)) B. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x + 1) / (cos x + 1)) C. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x - sin x + 1) / (cos x - sin x + 1)) D. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x sin x) / (cos x sin x)) Distribute in the denominator and combine like terms. Write the answer in terms of sine and cosine. (cos x - sin x + 1)(cos x + 1) / (cos^2 x + sin x + cos x - 1) (Do not factor.)

          Verify the identity. (cos x - sin x + 1) / (cos x + sin x - 1) = (cos x + 1) / sin x
To simplify the left side of the identity, multiply the numerator and denominator of the left side of the equation by the same factor. Choose the multiplication below that will be most useful in simplifying the left side of the equation.
A. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x + sin x - 1) / (cos x + sin x - 1))
B. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x + 1) / (cos x + 1))
C. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x - sin x + 1) / (cos x - sin x + 1))
D. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x sin x) / (cos x sin x))
Distribute in the denominator and combine like terms. Write the answer in terms of sine and cosine.
(cos x - sin x + 1)(cos x + 1) / (cos^2 x + sin x + cos x - 1) (Do not factor.)

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Verify the identity. (cos x - sin x + 1) / (cos x + sin x - 1) = (cos x + 1) / sin x
To simplify the left side of the identity, multiply the numerator and denominator of the left side of the equation by the same factor. Choose the multiplication below that will be most useful in simplifying the left side of the equation.
A. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x + sin x - 1) / (cos x + sin x - 1))
B. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x + 1) / (cos x + 1))
C. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x - sin x + 1) / (cos x - sin x + 1))
D. ((cos x - sin x + 1) / (cos x + sin x - 1)) * ((cos x sin x) / (cos x sin x))
Distribute in the denominator and combine like terms. Write the answer in terms of sine and cosine.
(cos x - sin x + 1)(cos x + 1) / (cos^2 x + sin x + cos x - 1) (Do not factor.)

Added by Tanya D.

Precalculus with Limits

Ron Larson 2nd Edition

Instant Answer

Solved by Expert Gregory Higby

01/28/2024

Step 1

This will help us eliminate the square root terms in the denominator. $$\frac{\cos x - \sin x + 1}{\cos x + \sin x - 1} \cdot \frac{\cos x - \sin x + 1}{\cos x - \sin x + 1}$$ Now, we can multiply the numerators and the denominators: $$\frac{(\cos x - \sin x + Show more…

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Verify the identity. (cos x - sin x + 1) / (cos x + sin x - 1) = (cos x + 1) / sin x To simplify the left side of the identity, multiply the numerator and denominator of the left side of the equation by the same factor. Choose the multiplication below that will be most useful in simplifying the left side of the equation. Distribute in the denominator and combine like terms. Write the answer in terms of sine and cosine. (cos x - sin x + 1)(cos x + 1) / (cos^2 x + sin x + cos x - 1) (Do not factor.)

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Transcript

00:01 I'm going at this problem with following the directions, i guess, that you have that cosine of x minus sine of x plus 1 times the other fraction, cosine of x plus 1, all over.

00:23 Let me just write out that it's being multiplied first.

00:26 I got to get this down first.

00:28 First, cosine of x plus sine of x plus 1 minus 1 times cosine of x plus 1.

00:40 Okay, so first of all, you have the numerator correct.

00:43 So let me just put a box around that and just say, make sure you leave your numerator alone.

00:47 But let's look at the denominator a little bit closer.

00:50 If you distribute, i agree with cosine squared.

00:53 But then you have to distribute again.

00:55 Cosine times 1 is just cosine.

00:58 But now i need to distribute the sine in here.

01:00 So that's sine of x cosine of x times here that's sine of x and then minus cosine of x because negative one times anything is just that thing but negative and then negative one times one is negative one so what i'm thinking about here is these cosines of x's cancel good they're gone but then from here, i'm thinking about what is equal to cosine squared of x.

01:34 And what i'm thinking about is how sine squared plus cosine squared equals one.

01:42 So what if i subtract sine squared over, then cosine squared of x equals one minus sine squared of x...

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