The expression you provided is:
\[ \cot^2 x \sec x - \cos x \]
We can start by rewriting the trigonometric identities involved:
- \(\cot x = \frac{\cos x}{\sin x}\), so \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\).
- \(\sec x = \frac{1}{\cos x}\).
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.
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