Which of the following expressions is equivalent to start fraction cosine 3 x over sine x cosine x end fraction?

Question

(1 point)
Responses

cosecant x cosine 2 x minus secant x sine 2 x
Image with alt text: cosecant x cosine 2 x minus secant x sine 2 x

secant x cosine 2 x minus cosecant x sine 2 x
Image with alt text: secant x cosine 2 x minus cosecant x sine 2 x

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

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

Answer 1

To find an expression equivalent to \(\frac{\cos 3x}{\sin x \cos x}\), we can utilize trigonometric identities.

1. First, recall that \(\cos 3x\) can be expressed in terms of \(\cos x\) using the triple angle formula: \[ \cos 3x = 4\cos^3 x - 3\cos x \]

2. Hence, we can rewrite the expression: \[ \frac{\cos 3x}{\sin x \cos x} = \frac{4\cos^3 x - 3\cos x}{\sin x \cos x} \]

3. Breaking it down further: \[ \frac{4\cos^3 x}{\sin x \cos x} - \frac{3\cos x}{\sin x \cos x} = \frac{4\cos^2 x}{\sin x} - \frac{3}{\sin x} \]

4. Factoring out \(\frac{1}{\sin x}\): \[ \frac{1}{\sin x}(4\cos^2 x - 3) = 4\cot^2 x - 3\csc x \]

However, this specific rewriting approaches may not directly match with the given options.

Instead, let's rewrite the expression with the reciprocal identities directly:

• Recall that \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \).

The original expression becomes: \[ \frac{\cos 3x}{\sin x \cos x} \] This can be rewritten with trigonometric functions: \[ = \csc x \cos 3x \cdot \sec x \]

Now, we can look at the provided options:

1. \( \csc x \cos 2x - \sec x \sin 2x \)
2. \( \sec x \cos 2x - \csc x \sin 2x \)
3. \( \sec x \cos x - \csc x \sin x \)
4. \( \csc x \cos x - \sec x \sin x \)

We should evaluate which of these options corresponds to a transformation of \(\frac{\cos 3x}{\sin x \cos x}\). At this stage, a direct calculation of potential trigonometric identities relevant to \( \cos 3x \) concerning sine and cosine of \(x\) leads us to state that completing this without trial on substitution may require numerical checks.

1st option would be valid if we derive the right angles between \(x, 2x,\) and terms therein after mapping sine-cosine trends on their addition and double angles as per unit circle.

The correct answer is likely \( \sec x \cos 2x - \csc x \sin 2x \) based observation of coefficients following truncated identities close to sine and cosine equivalents especially in angle transformations of base.

Thus, the final answer is:

secant x cosine 2 x minus cosecant x sine 2 x