Question

Question

A student was asked to prove cosine left-parenthesis x plus pi right-parenthesis equals negative cosine x.

The student’s work follows.

cosine left-parenthesis x plus pi right-parenthesis equals cosine x cosine pi plus sine x sine pi equals negative cosine x

Where was the student’s mistake?
(1 point)
Responses

The second expression should be cosine x sine pi plus sine x cosine pi.
The second expression should be Image with alt text: cosine x sine pi plus sine x cosine pi .

The second expression should be cosine x cosine pi minus sine x sine pi.
The second expression should be Image with alt text: cosine x cosine pi minus sine x sine pi .

The second expression should be cosine x sine pi minus sine x cosine pi.
The second expression should be Image with alt text: cosine x sine pi minus sine x cosine pi .

The second expression should be cosine x sine pi plus sine x sine pi.
The second expression should be Image with alt text: cosine x sine pi plus sine x sine pi .
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Answer 1

The student's work shows the application of the cosine addition formula incorrectly.

The correct cosine addition formula is:

\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]

In the student's work, they wrote:

\[ \cos(x + \pi) = \cos x \cos \pi + \sin x \sin \pi \]

Instead, it should be:

\[ \cos(x + \pi) = \cos x \cos \pi - \sin x \sin \pi \]

Given that \(\cos \pi = -1\) and \(\sin \pi = 0\), it simplifies to:

\[ \cos(x + \pi) = \cos x (-1) - \sin x (0) = -\cos x \]

Therefore, the error occurs in the student's expression for the cosine addition formula.

The correct response is:

The second expression should be cosine x cosine pi minus sine x sine pi.