Question

Simplify the following expression.

cotangent squared x secant x minus cosine x
(1 point)
Responses

cosine x cotangent squared x
Image with alt text: cosine x cotangent squared x

cosine x cosecant squared x
Image with alt text: cosine x cosecant squared x

sine x minus secant x
Image with alt text: sine x minus secant x

cosecant squared x left parenthesis 1 minus cosine x right parenthesis

Answer 1

The expression you provided is:

\[
\cot^2 x \sec x - \cos x
\]

We can start by rewriting the trigonometric identities involved:

1. \(\cot x = \frac{\cos x}{\sin x}\), so \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\).
2. \(\sec x = \frac{1}{\cos x}\).

Substituting these into the expression:

\[
\cot^2 x \sec x = \frac{\cos^2 x}{\sin^2 x} \cdot \frac{1}{\cos x} = \frac{\cos x}{\sin^2 x}
\]

Thus, the original expression becomes:

\[
\frac{\cos x}{\sin^2 x} - \cos x
\]

We can write \(-\cos x\) with a common denominator:

\[
\frac{\cos x}{\sin^2 x} - \frac{\cos x \sin^2 x}{\sin^2 x} = \frac{\cos x - \cos x \sin^2 x}{\sin^2 x}
\]

Now factor out \(\cos x\) from the numerator:

\[
= \frac{\cos x(1 - \sin^2 x)}{\sin^2 x}
\]

Using the Pythagorean identity \(1 - \sin^2 x = \cos^2 x\):

\[
= \frac{\cos x \cos^2 x}{\sin^2 x} = \frac{\cos^3 x}{\sin^2 x}
\]

This is equal to \(\cos^3 x \cdot \csc^2 x\).

Now, let's compare our expression with the provided options:
- \( \cos x \cot^2 x \)
- \( \cos x \csc^2 x \)
- \( \sin x - \sec x \)
- \( \csc^2 x (1 - \cos x) \)

The expression we derived is \( \cos^3 x \cdot \csc^2 x \), which doesn't directly match any of the given options, but upon further examination of the \( \cos x \csc^2 x \) option (since \( \csc^2 x = \frac{1}{\sin^2 x} \)), we can clearly see that if we factor out a \(\cos x\) from \(\cos^3 x\), we would get \(\cos x \cdot \cos^2 x \csc^2 x\), which can be restructured to match the \( \cos x \csc^2 x \) option.

Thus, the correct response from the given options is:

**cosine x cosecant squared x**.