Now, $ t = \tan x + \cot x \geq 2 $ by AM-GM, with equality when $ \tan x = 1 \Rightarrow x = \frac{\pi}{4} $.

["Title: Prove $ \ an x + \cot x \geq 2 $ Using AM-GM Inequality (Equality at $ x = \frac{\pi}{4} $)", "Meta Description:\nDiscover how to prove $ \ an x + \cot x \geq 2 $ for all real $ x $ where $ \ an x $ and $ \cot x $ defined, using the AM-GM inequality. Learn when equality holds, specifically when $ x = \frac{\pi}{4} $.", "---", "## Proving $ \ an x + \cot x \geq 2 $ by AM-GM Inequality", "Mathematics often reveals elegant truths through inequality proofs, and one of the simplest yet powerful demonstrations is proving the inequality:", "$$\n\ an x + \cot x \geq 2\n$$", "This inequality holds for all real $ x $ where $ \ an x > 0 $ and $ \cot x > 0 $, i.e., in the intervals $ x \equiv \frac{\pi}{4} \pmod{\frac{\pi}{2}} $, excluding where either function is undefined or zero. In such valid domains, the Arithmetic Mean–Geometric Mean (AM-GM) Inequality provides a clear, intuitive path to the solution.", "---", "### Step 1: Understanding the Expression", "We begin with the identity:\n$$\n\ an x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}\n$$\nThus:\n$$\n\ an x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\n$$", "Combining the fractions:\n$$\n\ an x + \cot x = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x}\n$$", "Using the Pythagorean identity $ \sin^2 x + \cos^2 x = 1 $:\n$$\n\ an x + \cot x = \frac{1}{\sin x \cos x}\n$$", "So the inequality becomes:\n$$\n\frac{1}{\sin x \cos x} \geq 2\n\quad \Rightarrow \quad \sin x \cos x \leq \frac{1}{2}\n$$", "---", "### Step 2: Applying AM-GM Inequality", "We now apply the AM-GM inequality to $ \sin^2 x $ and $ \cos^2 x $, which are always non-negative and positive in valid domains (i.e., $ x $ not an integer multiple of $ \frac{\pi}{2} $):", "By AM-GM:\n$$\n\frac{\sin^2 x + \cos^2 x}{2} \geq \sqrt{\sin^2 x \cdot \cos^2 x}\n\quad \Rightarrow \quad \frac{1}{2} \geq |\sin x \cos x|\n$$", "Since $ \sin x \cos x \leq \frac{1}{2} $, and both functions can be positive or negative depending on $ x $, but for $ \ an x + \cot x $ to be defined and real, $ \ an x > 0 $ and $ \cot x > 0 $ imply $ \sin x \cos x > 0 $, so:\n$$\n\sin x \cos x \leq \frac{1}{2}\n$$", "But from AM-GM:\n$$\n\sin^2 x + \cos^2 x = 1 \geq 2 |\sin x \cos x| \Rightarrow |\sin x \cos x| \leq \frac{1}{2}\n$$", "Thus,\n$$\n\frac{1}{\sin x \cos x} \geq \frac{1}{\frac{1}{2}} = 2\n$$\nHowever, note that $ \sin x \cos x $ can be negative in some intervals, but in regions where $ \ an x + \cot x $ is defined and positive, $ \sin x \cos x > 0 $. Therefore, focusing on intervals where $ \sin x \cos x > 0 $, we get:\n$$\n\ an x + \cot x = \frac{1}{\sin x \cos x} \geq 2\n$$", "---", "### Step 3: Equality Condition", "Equality in AM-GM holds when the two numbers are equal:\n$$\n\sin^2 x = \cos^2 x \Rightarrow \sin^2 x = \frac{1}{2} \Rightarrow \sin x = \pm \frac{\sqrt{2}}{2}, \quad \cos x = \pm \frac{\sqrt{2}}{2}\n$$", "At $ x = \frac{\pi}{4} + n\frac{\pi}{2} $, specifically $ x = \frac{\pi}{4} $ in the principal interval $ \left(0, \frac{\pi}{2}\right) $, both $ \sin x = \cos x = \frac{\sqrt{2}}{2} $, so:\n$$\n\ an x + \cot x = 1 + 1 = 2\n$$", "This confirms $ \ an x + \cot x \geq 2 $ with equality if and only if $ \ an x = 1 $, i.e., $ x = \frac{\pi}{4} + n\pi $ for integer $ n $. But within the domain where $ \ an x > 0 $ and $ \cot x > 0 $, the equality holds uniquely (modulo periodicity) at $ x = \frac{\pi}{4} $.", "---", "### Conclusion", "The AM-GM inequality elegantly proves the fundamental inequality:\n$$\n\ an x + \cot x \geq 2 \quad \ ext{for all } x \ ext{ where defined and } \ an x, \cot x > 0\n$$\nwith equality achieved precisely when $ \ an x = 1 $, i.e., $ x = \frac{\pi}{4} + n\pi $, $ n \in \mathbb{Z} $.", "This result not only simplifies trigonometric inequalities but also highlights the deep connection between algebraic inequalities and geometric identities in mathematics.", "---", "Keywords: $ \ an x + \cot x \geq 2 $, AM-GM inequality, $ \sin x \cos x $, equality condition $ x = \frac{\pi}{4} $, $ \ an x = 1 $, inequality proof, trigonometric inequality.", "Read more:\n- Prove $ \sin x + \cos x \leq \sqrt{2} $ using trigonometric identities\n- Apply AM-GM to other trigonometric expressions\n- Master key inequalities in trigonometry with AM-GM", "---", "@inqmath — Mastering inequalities starts with understanding symmetry and positivity—never underestimate AM-GM!"]