Simplify the following expression: \( \cos 7 x \cos 3 x+\sin 7 x \sin 3 x \)

Did you know that the expression you've provided can actually be simplified using a trigonometric identity? The cosine of the difference of angles is given by the formula: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] So, if we let \( A = 7x \) and \( B = 3x \), we can rewrite your expression as: \[ \cos(7x - 3x) = \cos(4x) \] Now it becomes clear that \( \cos 7 x \cos 3 x + \sin 7 x \sin 3 x = \cos(4x) \). Isn’t that neat? This identity is not just a mathematical trick; it has real-world applications in fields like physics and engineering. For example, when analyzing oscillations or wave patterns, being able to simplify expressions involving sines and cosines can make it easier to compute frequencies, amplitudes, and phase shifts. By recognizing these identities, you can save time and simplify complex problems when dealing with waves or harmonic motion!