Simplify the following expression.

Question

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.