ight) = 1 - rac{1}{2}(\cos rac{\pi}{2} - \cos \pi) = 1 - rac{1}{2}(0 - (-1)) = 1 - rac{1}{2}(1) = 0.5 $. Correct.

["Mastering Trigonometric Simplification: A Step-by-Step Breakdown of the Expression", "Understanding trigonometric identities can sometimes feel like solving a puzzle—especially when dealing with complex expressions involving cosine evaluations. Today, we break down one of these expressions step by step to reveal its elegant simplicity:", "[\night) = 1 - \frac{1}{2}\left( \cos \frac{\pi}{2} - \cos \pi \right) = 1 - \frac{1}{2}(0 - (-1)) = 1 - \frac{1}{2}(1) = 0.5\n]", "### Step 1: Evaluate the Cosine Values", "Start by determining the cosine values of the key angles:", "- ( \cos \frac{\pi}{2} = 0 ) — This is a standard angle on the unit circle, where cosine equals zero.\n- ( \cos \pi = -1 ) — At ( \pi ) radians (180 degrees), cosine reaches its minimum value of -1.", "Substitute these values into the original expression:", "[\night) = 1 - \frac{1}{2}\left( 0 - (-1) \right)\n]", "### Step 2: Simplify the Parentheses", "Inside the parentheses, simplify the subtraction:", "[\n0 - (-1) = 0 + 1 = 1\n]", "Now the expression becomes:", "[\night) = 1 - \frac{1}{2}(1)\n]", "### Step 3: Perform the Final Multiplication and Subtraction", "Multiply ( \frac{1}{2} \ imes 1 = \frac{1}{2} ), then subtract:", "[\n1 - \frac{1}{2} = \frac{2}{2} - \frac{1}{2} = \frac{1}{2} = 0.5\n]", "### Why This Simplification Matters", "This example illustrates the power of replacing trigonometric constants with their exact values before simplifying algebraically. Recognizing that:", "- ( \cos \frac{\pi}{2} = 0 )\n- ( \cos \pi = -1 )", "allows for accurate and efficient calculation. These foundational values frequently appear in calculus, physics, engineering, and computer graphics—making mastery of such simplifications essential for students, professionals, and anyone studying mathematical functions.", "### Final Takeaway", "When simplifying trigonometric expressions:", "- Know key cosine values at standard angles.\n- Simplify inside parentheses first.\n- Perform operations step by step.", "Understanding this expression not only confirms that\n[\night) = 0.5\n]\nbut also builds confidence in manipulating trigonometric functions under any context.", "Key Terms: trigonometric simplification, cosine values, algebraic simplification, unit circle, exact trigonometric values\nKeywords: cos(π/2), cos(π), trigonometric identity, mathematical simplification, exact calculation"]