Solution: Expand $ (\tan x + \cot x)^2 = \tan^2 x + 2 \tan x \cot x + \cot^2 x $. Simplify using $ \tan x \cot x = 1 $: $ \tan^2 x + 2 + \cot^2 x $. Use identity $ \tan^2 x + \cot^2 x = (\tan x - \cot x)^2 + 2 $, so expression becomes $ (\tan x - \cot x)^2 + 4 $. The minimum occurs when $ \tan x = \cot x $, i.e., $ x = \frac{\pi}{4} + k\frac{\pi}{2} $, giving $ 0 + 4 = 4 $. \boxed{4}
["Simplifying $ (\ an x + \cot x)^2 $: A Powerful Trigonometric Identity & Minimum Value", "Trigonometric identities are powerful tools that simplify complex expressions and reveal hidden patterns in mathematics. One such elegant identity involves the sum $ (\ an x + \cot x)^2 $, which, when expanded and simplified using fundamental relationships, reveals a minimal value of just 4.", "---", "### Step 1: Expand the Square", "Start with the expression:", "$$\n(\ an x + \cot x)^2\n$$", "Using the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$, we expand:", "$$\n(\ an x + \cot x)^2 = \ an^2 x + 2 \ an x \cot x + \cot^2 x\n$$", "---", "### Step 2: Use the Identity $ \ an x \cot x = 1 $", "Since both $\ an x$ and $\cot x$ are reciprocals, their product is 1:", "$$\n2 \ an x \cot x = 2 \cdot 1 = 2\n$$", "Substitute back:", "$$\n(\ an x + \cot x)^2 = \ an^2 x + 2 + \cot^2 x\n$$", "---", "### Step 3: Apply a Key Trigonometric Identity", "We now use a powerful identity involving squares:", "$$\n\ an^2 x + \cot^2 x = (\ an x - \cot x)^2 + 2\n$$", "This identity shows that $ \ an^2 x + \cot^2 x $ is always at least 2, since a square is non-negative:", "$$\n(\ an x - \cot x)^2 \geq 0 \Rightarrow \ an^2 x + \cot^2 x \geq 2\n$$", "---", "### Step 4: Substitute Back to Simplify the Full Expression", "Recall from Step 2:", "$$\n(\ an x + \cot x)^2 = \ an^2 x + 2 + \cot^2 x\n$$", "Now substitute the minimum value:", "$$\n= (\ an^2 x + \cot^2 x) + 2 \geq 2 + 2 = 4\n$$", "---", "### Step 5: Determine When the Minimum Occurs", "The minimum value of 4 occurs when:", "$$\n\ an x - \cot x = 0 \Rightarrow \ an x = \cot x\n$$", "Given $ \cot x = \frac{1}{\ an x} $, this implies:", "$$\n\ an x = \frac{1}{\ an x} \Rightarrow \ an^2 x = 1 \Rightarrow \ an x = \pm 1\n$$", "So:", "$$\nx = \frac{\pi}{4} + k\frac{\pi}{2}, \quad k \in \mathbb{Z}\n$$", "At these angles, the original expression reaches its minimum.", "---", "### Final Answer: The Minimum Value is $$\n\boxed{4}\n$$", "This elegant derivation shows how algebraic manipulation combined with trigonometric identities leads to clear and powerful simplifications—essential skills in calculus, physics, and engineering applications.", "---", "Simplify $ (\ an x + \cot x)^2 $ efficiently and discover its minimum—easier than you think!"]
