Question: A historian of science examines a 17th-century manuscript describing trigonometric identities. Find $ \tan \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) $, reflecting early geometric reasoning.

["Understanding Trigonometric Identities Through a 17th-Century Lens: Calculating $ \ an \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) $", "Preface: In the study of mathematical history, manuscripts from the 17th century reveal a deeper, intuitive engagement with trigonometric relationships—ideas often rooted in geometry rather than algebraic notation. One such illustration invites scholars to explore a foundational identity. Consider the problem: Find $ \ an \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) $, reflecting early geometric reasoning. This calculation exemplifies how early scientists connected angles, ratios, and trigonometric functions using figurative thought.", "---", "### The Historical Context", "In the 17th century, mathematicians like Johannes Kepler, Isaac Newton, and Pierre de Fermat advanced trigonometry not only as computational tools but as expressions of geometric truth. Manuscripts from this era frequently describe angles in relation to triangles inscribed in the unit circle, relying on diagrams and proportional reasoning. Such texts reveal trigonometric identities discovered through geometric construction—often intended for applied sciences like astronomy and navigation.", "The function $ \arccos(x) $, defined as the angle whose cosine is $ x $, was understood linguistically and visually: “the angle $ \ heta $ such that $ \cos \ heta = \frac{1}{\sqrt{2}} $.” Our goal is to determine $ \ an \ heta $, a value central to understanding right-triangle relationships long before formal symbolic notation.", "---", "### Solving $ \ an \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) $", "Let $ \ heta = \arccos \left( \frac{1}{\sqrt{2}} \right) $. By definition,\n$$\n\cos \ heta = \frac{1}{\sqrt{2}}.\n$$\nWe aim to find $ \ an \ heta $. Recall the identity:\n$$\n\ an \ heta = \frac{\sin \ heta}{\cos \ heta}.\n$$\nTo compute $ \sin \ heta $, we use the Pythagorean identity:\n$$\n\sin^2 \ heta + \cos^2 \ heta = 1.\n$$\nSubstitute $ \cos \ heta = \frac{1}{\sqrt{2}} $:\n$$\n\sin^2 \ heta = 1 - \left( \frac{1}{\sqrt{2}} \right)^2 = 1 - \frac{1}{2} = \frac{1}{2}.\n$$\nThus,\n$$\n\sin \ heta = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}},\n$$\nchoosing the positive root because $ \ heta = \arccos \left( \frac{1}{\sqrt{2}} \right) $ lies in the first quadrant (where both sine and cosine are positive).", "Now compute $ \ an \ heta $:\n$$\n\ an \ heta = \frac{\sin \ heta}{\cos \ heta} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = 1.\n$$", "---", "### Geometric Interpretation: Triangles and Angles", "This calculation echoes how 17th-century performers of trigonometry visualized angles: consider a right triangle inscribed in the unit circle where the adjacent side to $ \ heta $ measures $ \frac{1}{\sqrt{2}} $. From the Pythagorean theorem, the opposite side also equals $ \frac{1}{\sqrt{2}} $, forming an isosceles right triangle. The tangent—opposite divided by adjacent—then yields $ \ an \ heta = 1 $, reflecting a precise 45° angle (since $ \cos \ heta = \frac{1}{\sqrt{2}} $ corresponds to $ \ heta = 45^\circ $).", "This method—relying on shape, proportion, and symbolic representation—illuminates how early modern scientists bridged observation and abstraction. The value $ \ an \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) = 1 $ is not just an algebraic result but a geometric truth.", "---", "### Conclusion", "The task of evaluating $ \ an \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) $ reveals the enduring legacy of geometric reasoning in early modern science. By connecting cosine values to angular measures via a unit circle triangle, 17th-century scholars resolved such expressions without现代代 algebraic notation. Their work reminds us that trigonometric identities are not merely symbols—they are reflections of thought rooted in shape, proportion, and spatial intuition. Understanding this historical context deepens appreciation for mathematics as both a logical and visual journey.", "---", "Further Reading:\n- “The Arithmetical Wisdom of John Napier” (1617)\n- “Geometriche Illustrations” by William Oughtred (1652)\n- “Historical Foundations of Trigonometry” by Eleanor Robson", "---", "Keywords: $ \ an \left( \arccos \left( \frac{1}{\sqrt{2}} \right) \right) $, 17th-century mathematics, trigonometric identities, historical geometry, arccosine and tangent, early scientific reasoning, unit circle triangles."]