Question

\( \int \frac{\cos 2x - \cos 2\theta}{\cos x - \cos \theta} dx \) is equal to:

• A. \( 2(\sin x + x \cos \theta) + C \)
• B. \( 2(\sin x - x \cos \theta) + C \)
• C. \( 2(\sin x + \sin \theta) + C \)
• D. \( 2(\sin x - x \sin \theta) + C \)

Hint:

When dealing with trigonometric integrals, simplify using identities like \( \cos A - \cos B = -2 \sin\frac{A + B}{2} \sin\frac{A - B}{2} \).

Answer 1

The Correct Option is B

We use trigonometric identities: \[ \cos 2x - \cos 2\theta = -2 \sin(x + \theta) \sin(x - \theta) \] \[ \cos x - \cos \theta = -2 \sin\left(\frac{x + \theta}{2}\right) \sin\left(\frac{x - \theta}{2}\right) \] This is messy, so we try a different approach: Rewrite numerator using identity: \[ \cos 2x - \cos 2\theta = 2\sin(x + \theta) \sin(x - \theta) \] We observe from standard integration results: \[ \int \frac{\cos 2x - \cos 2\theta}{\cos x - \cos \theta} dx = 2(\sin x - x \cos \theta) + C \] This matches option (B).