Trigonometry & Inverse Trigonometry
Trigonometric Expressions and Simplification
Grade 11
Question:
<p>The minimum value of the function <span>f(x) = </span><span>sin x</span>/<span>√(1 - cos² x)</span> + <span>cos x</span>/<span>√(1 - sin² x)</span> + <span>tan x</span>/<span>√(sec² x - 1)</span> + <span>cot x</span>/<span>√(cosec² x - 1)</span> whenever it is defined is</p>
<p>(a) 4</p>
<p>(b) 2</p>
<p>(c) 0</p>
<p>(d) -2</p>
Step-by-Step Solution
Key Concept: Recognize that each fraction simplifies to ±1 depending on the quadrant, then find when their sum is minimized.
<p><strong>Step 1:</strong> Simplify each term:</p><p>sin x / √(1 - cos² x) = sin x / |sin x| = ±1</p><p>cos x / √(1 - sin² x) = cos x / |cos x| = ±1</p><p>tan x / √(sec² x - 1) = tan x / |tan x| = ±1</p><p>cot x / √(cosec² x - 1) = cot x / |cot x| = ±1</p><p><strong>Step 2:</strong> Each term equals ±1 depending on the sign of the trigonometric function in that quadrant.</p><p><strong>Step 3:</strong> The minimum value occurs when the sum is minimized. Since each term is ±1, the minimum is achieved when terms are negative.</p><p>∴ Answer is (b) 2</p>
Correct Answer: B