\tan 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}.

["# Understanding the Tangent of 75°: A Step-by-Step Derivation", "The tangent of 75 degrees is a classic trigonometric expression that frequently appears in geometry, calculus, and advanced algebra. While many memorize values like ( \ an 75^\circ = 2 + \sqrt{3} ), understanding how this result is derived provides deeper insight into trigonometric identities and rationalizing techniques. In this article, we break down the elegant derivation of\n[\n\ an 75^\circ = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\n]\nusing algebraic manipulation and fundamental identities.", "## The Identity Behind the Equation", "To express ( \ an 75^\circ ) in fraction form, we start with the well-known tangent addition formula:\n[\n\ an(A + B) = \frac{\ an A + \ an B}{1 - \ an A \ an B}\n]\nSince ( 75^\circ = 45^\circ + 30^\circ ), we apply this identity with ( A = 45^\circ ) and ( B = 30^\circ ).", "We know:\n- ( \ an 45^\circ = 1 )\n- ( \ an 30^\circ = \frac{1}{\sqrt{3}} )", "Substituting into the formula:\n[\n\ an 75^\circ = \frac{\ an 45^\circ + \ an 30^\circ}{1 - \ an 45^\circ \ an 30^\circ} = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}}\n]", "This confirms the starting expression. The next step involves simplifying the rational expressions to their preferred form.", "## Simplifying the Expression", "We begin with the numerator and denominator of the current fraction:\n[\n\frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}}\n]\nTo eliminate the square roots in the denominators and create a unified structure, multiply both numerator and denominator by ( \sqrt{3} ):\n[\n\frac{\left(1 + \frac{1}{\sqrt{3}}\right) \cdot \sqrt{3}}{\left(1 - \frac{1}{\sqrt{3}}\right) \cdot \sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\n]", "This matches the target expression, demonstrating how rationalization transforms an intermediate form into a cleaner algebraic representation.", "## Rationalizing and Simplifying Further", "While the current form ( \frac{\sqrt{3} + 1}{\sqrt{3} - 1} ) is mathematically valid, it can be rationalized to eliminate the radical in the denominator—often preferred in calculus, algebra, and textbook solutions. Rationalization involves multiplying numerator and denominator by the conjugate ( \sqrt{3} + 1 ):\n[\n\frac{\sqrt{3} + 1}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - (1)^2}\n]\nCompute numerator and denominator:\n- Numerator: ( (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} )\n- Denominator: ( 3 - 1 = 2 )", "So:\n[\n\frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}\n]", "This confirms the well-known exact value:\n[\n\ an 75^\circ = 2 + \sqrt{3} \approx 3.732\n]", "## Why This Derivation Matters", "Understanding how to simplify ( \ an 75^\circ ) from its angle-addition form to different algebraic forms enhances problem-solving flexibility. The expression ( \frac{\sqrt{3} + 1}{\sqrt{3} - 1} ) is particularly useful when analyzing asymptotic behavior in calculus, solving trigonometric equations, or preparing for identity-based proofs.", "Moreover, mastering rationalization techniques ensures readiness for higher-level math applications where standardized forms improve clarity and reduce computational errors.", "## Conclusion", "The journey from ( \ an 75^\circ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} ) to ( \frac{\sqrt{3} + 1}{\sqrt{3} - 1} ) highlights the power of substitution and algebraic manipulation in trigonometry. While multiple forms exist, recognizing when to rationalize or simplify supports deeper comprehension and practical problem-solving. Whether memorizing exact values or exploring derivative properties, grasping these transformations strengthens your mathematical toolkit.", "Explore broader applications of trigonometric identities in geometry, physics, and engineering—mastery begins with understanding expressions like those of ( \ an 75^\circ ).", "---", "### Keywords:\ntan 75°, trigonometric identities, tangent addition formula, rationalizing denominators, √3 in trig, exact value tan 75°, 75 degree tangent derivation, algebraic simplification, calculus prep, trigonometric forms."]