If (\cos \theta-\cos(-\theta)=q), what is the value of (q)?

Correct answer: (0)

The direct answer is option A: \(q=0\). To solve this, remember that cosine is an even function. An even function has the same value when its input changes from \(\theta\) to \(-\theta\), so \(\cos(-\theta)=\cos\theta\). Therefore, \(q=\cos\theta-\cos(-\theta)=\cos\theta-\cos\theta=0\). Option A is correct because the two equal quantities cancel. Option B, \(2\cos\theta\), would result from adding the two cosine terms, not subtracting equal terms. Option C, \(-2\cos\theta\), also does not follow from the given subtraction. Option D, 1, is not generally the value of this difference; cosine may have many values depending on \(\theta\). The key exam cue is: cosine is even, so its negative-angle value is unchanged. Do not confuse this with sine, because sine is an odd function and \(\sin(-\theta)=-\sin\theta\).