Solution: Let $ \theta = \arccos\left(\frac{1}{\sqrt{2}}\right) $, so $ \cos \theta = \frac{1}{\sqrt{2}} $. Then $ \theta = 45^\circ $, and $ \tan \theta = 1 $. The answer is $ \boxed{1} $.
["Understanding the Solution: Let $ \ heta = \arccos\left(\frac{1}{\sqrt{2}}\right) $ — A Step-by-Step Guide", "In trigonometry, solving for key angles and their corresponding ratios is essential for mastering identities and equations. One elegant mathematical solution arises when we set\n$$\n\ heta = \arccos\left(\frac{1}{\sqrt{2}}\right)\n$$\nThis equality tells us that $ \cos \ heta = \frac{1}{\sqrt{2}} $. Recognizing this value connects directly to a well-known angle in the unit circle, unlocking valuable trigonometric insights.", "Since $ \cos 45^\circ = \frac{1}{\sqrt{2}} $, it follows that\n$$\n\ heta = 45^\circ = \frac{\pi}{4} \ ext{ radians}\n$$\nFrom here, we easily determine $ \ an \ heta $:\n$$\n\ an \ heta = \ an(45^\circ) = 1\n$$\nThis result simplifies cleanly to the value $ 1 $, showing a direct relationship between cosine, tangent, and fundamental angle measures.", "The boxed answer,\n$$\n\boxed{1}\n$$\nis not just a number—it reflects the harmony between trigonometric functions and geometric angle measures.", "Why This Matters\nUnderstanding how specific cosine values lead to exact tangent results strengthens problem-solving skills in identities, equation solving, and calculus applications. It also illustrates how radians and degrees complement each other in trigonometric calculations.", "Whether you're a student exploring trigonometry basics or a perceptive learner appreciating mathematical elegance, recognizing $ \ heta = \arccos\left(\frac{1}{\sqrt{2}}\right) = 45^\circ $ and $ \ an \ heta = 1 $ offers clarity, precision, and confidence in advanced math.", "Key Takeaways\n- $ \arccos\left(\frac{1}{\sqrt{2}}\right) = 45^\circ $\n- $ \ an(45^\circ) = 1 $\n- The solution exemplifies the interplay between inverse and direct trigonometric functions\n- Simplifying expressions like $ \cos \ heta = \frac{1}{\sqrt{2}} $ builds a strong foundation for advanced topics in algebra and calculus", "Embrace these relationships— they make complex math accessible and intuitive."]
