Question: Find all angles $ z \in [0^\circ, 360^\circ] $ such that $ \tan z + \cot z = 2 $.

["Title: Solve $ \ an z + \cot z = 2 $: Find All Solutions in $[0^\circ, 360^\circ]$", "Meta Description:\nLearn how to solve $ \ an z + \cot z = 2 $ and find all angles $ z \in [0^\circ, 360^\circ] $ that satisfy this equation. Step-by-step explanation with exact solutions and key insights.", "---", "## Finding All Angles $ z \in [0^\circ, 360^\circ] $ Such That $ \ an z + \cot z = 2 $", "Trigonometric equations can reveal elegant symmetry and specific angle solutions, especially when dealing with reciprocal functions like tangent and cotangent. One such equation is:", "$$\n\ an z + \cot z = 2\n$$", "For real values of $ z $, this equation has exact solutions that lie within the interval $[0^\circ, 360^\circ]$. In this article, we will uncover the general structure of the equation, derive the solutions, and explain why only certain angles satisfy the condition.", "---", "### Step 1: Rewrite the Equation Using Tangent Identity", "Recall that $ \cot z = \frac{1}{\ an z} $. Let $ x = \ an z $. Then the equation becomes:", "$$\nx + \frac{1}{x} = 2\n$$", "Multiply both sides by $ x $ (noting $ x <br/>\ne 0 $, since $ \ an z = 0 $ would lead to $ \cot z $ undefined):", "$$\nx^2 + 1 = 2x\n$$", "Bring all terms to one side:", "$$\nx^2 - 2x + 1 = 0\n$$", "This factors nicely:", "$$\n(x - 1)^2 = 0\n$$", "So, $ x = 1 $, which implies $ \ an z = 1 $.", "---", "### Step 2: Solve $ \ an z = 1 $ in the Interval $[0^\circ, 360^\circ]$", "The tangent function equals 1 at standard angles where sine and cosine are equal and non-zero. Recall:", "$$\n\ an z = \frac{\sin z}{\cos z} = 1 \implies \sin z = \cos z\n$$", "This occurs when $ z = 45^\circ + 180^\circ k $, where $ k $ is any integer. Within $[0^\circ, 360^\circ]$, the solutions are:", "- $ z = 45^\circ $\n- $ z = 45^\circ + 180^\circ = 225^\circ $", "Thus, the possible solutions are $ z = 45^\circ $ and $ z = 225^\circ $.", "---", "### Step 3: Verify Solutions in the Original Equation", "Before concluding, verify both values to ensure they are valid and avoid undefined cases.", "- For $ z = 45^\circ $:\n $$\n \ an 45^\circ = 1, \quad \cot 45^\circ = 1 \implies \ an z + \cot z = 1 + 1 = 2 \quad \checkmark\n $$", "- For $ z = 225^\circ $:\n $ \ an 225^\circ = \ an(180^\circ + 45^\circ) = \ an 45^\circ = 1 $, and similarly $ \cot z = 1 $, so again\n $$\n 1 + 1 = 2 \quad \checkmark\n $$", "Both values satisfy the equation.", "---", "### Step 4: Why Only $ \ an z = 1 $?", "Note that $ \cot z $ is undefined when $ \sin z = 0 $ (i.e., $ z = 0^\circ, 180^\circ, 360^\circ $), and $ \ an z $ undefined at $ z = 90^\circ, 270^\circ $. Since $ \ an z + \cot z $ is undefined at those points, they cannot be solutions.", "The equation $ \ an z + \cot z = 2 $ implies both functions are defined and real, so $ z $ must avoid asymptotes — confirming we only accept isolated points where $ \ an z = 1 $.", "---", "### Conclusion", "The complete solution set for $ \ an z + \cot z = 2 $ in the interval $[0^\circ, 360^\circ]$ is:", "$$\n\boxed{z = 45^\circ \quad \ ext{and} \quad z = 225^\circ}\n$$", "These angles reflect the symmetry and periodic nature of tangent and cotangent. Mastering this type of equation helps deepen understanding of trigonometric identities and function behavior across the unit circle.", "---", "## Additional Tips", "- Use the identity $ \ an z + \cot z = \frac{2}{\sin 2z} $ to rewrite the equation:\n $$\n \frac{2}{\sin 2z} = 2 \implies \sin 2z = 1\n $$\n Solving $ \sin 2z = 1 $ yields $ 2z = 90^\circ + 360^\circ k $, so $ z = 45^\circ + 180^\circ k $ — confirming our earlier result within $[0^\circ, 360^\circ]$.", "- Graphically, this corresponds to the points where the hyperbola $ y = \ an z + \cot z $ touches the line $ y = 2 $, confirming exact solutions.", "This elegant solution not only solves the equation cleanly but also illustrates the power of trigonometric identities and periodicity in finding precise angle measures.", "---", "Keywords:\nfind all angles $ z \in [0^\circ, 360^\circ] $, $ \ an z + \cot z = 2 $, solve $ \ an z + \cot z = 2 $, trigonometric equation solutions, $ \ an z = 1 $, triangle identities, unit circle applications.", "---", "Related Reading:\n- Solve $ \ an z = \cot z $\n- All solutions to $ \sin 2z = 1 $\n- Understanding reciprocal trigonometric functions", "---", "By mastering equations like $ \ an z + \cot z = 2 $, learners unlock deeper analytical skills essential for advanced trigonometry and mathematical problem solving."]