Solution: The tangent of $ 45^\circ $ is defined as $ \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $.
["The Tangent of 45°: A Fundamental Trigonometric Identity Explained", "Understanding trigonometric functions is essential for mastering geometry, physics, and engineering. One of the most foundational identities in trigonometry is the tangent of 45°, which reveals a key truth about right triangle relationships and the unit circle. The tangent of 45° is defined as:", "$$\n\ an 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\n$$", "### What Does This Mean?", "At first glance, this result may seem simple, but it opens the door to understanding deeper mathematical principles. The tangent function is defined as the ratio of the sine to the cosine of an angle. For 45°, both sine and cosine values are equal:", "- $ \sin 45^\circ = \frac{\sqrt{2}}{2} $\n- $ \cos 45^\circ = \frac{\sqrt{2}}{2} $", "Since these two values are identical, their ratio simply reduces to:", "$$\n\ an 45^\circ = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\n$$", "### The Geometric Intuition Behind Tangent = 1", "In a right triangle with a 45° angle, the two non-right sides (legs) are congruent—this is the defining property of a 45–45–90 triangle. Let each leg measure $ 1 $ unit. Using the Pythagorean theorem, the hypotenuse comes out to $ \sqrt{1^2 + 1^2} = \sqrt{2} $.", "Now compute tangent:", "- Opposite side = $ 1 $\n- Adjacent side = $ 1 $\n- $ \ an 45^\circ = \frac{1}{1} = 1 $", "This geometric interpretation reinforces the algebraic identity and illustrates why tangent at 45° is always 1—when the opposite and adjacent sides are equal.", "### Why Understanding This Matters", "The identity $ \ an 45^\circ = 1 $ is more than a textbook formula. It helps students:", "- Simplify complex trigonometric expressions\n- Recognize properties of special triangles\n- Apply right triangle relationships in real-world problems\n- Build confidence in solving equations involving angles", "Moreover, this basic identity supports advanced topics such as slope in coordinate geometry, where a 45° angle corresponds to a line with slope 1—further linking trigonometry to algebra.", "### Summary", "The tangent of 45° equals 1 because both the sine and cosine of 45° are $ \frac{\sqrt{2}}{2} $, and their ratio cancels perfectly. This elegant result stems from pure geometry and serves as a cornerstone in trigonometry. Whether you're calculating angles in a building design, analyzing waves in physics, or solving calculus problems, knowing that $ \ an 45^\circ = 1$ helps you unlock deeper mathematical insight.", "Keywords: tangent of 45 degrees, trigonometric identity, $ \ an 45^\circ $, sin 45°, cos 45°, unit circle, right triangle, $ \frac{\sqrt{2}}{2} $, geometry, trigonometry basics, slope in coordinate geometry."]
