<p><strong>Statement I:</strong> \(y = \tan^{-1}(\tan x)\) and \(y = \cos^{-1}(\cos x)\) are not the same function</p><br/><p><strong>Statement II:</strong> The range of \(\tan^{-1}(\tan x)\) is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) and the range of \(\cos^{-1}(\cos x)\) is \([0, \pi]\)</p>
<p>(a) Statement I is True; Statement II is True; Statement II is a correct explanation for Statement I</p>
<p>(b) Statement I is True; Statement II is True; Statement II is NOT a correct explanation for Statement I</p>
<p>(c) Statement I is True; Statement II is False</p>
<p>(d) Statement I is False; Statement II is True</p>
Step-by-Step Solution
Key Concept: Inverse trigonometric functions have restricted ranges, so tan⁻¹(tan x) and cos⁻¹(cos x) don't simply equal x for all real x. The key is understanding that tan⁻¹ and cos⁻¹ output values only within their defined ranges, which causes different behaviors on different intervals.
<p><strong>Step 1: Analyze y = tan⁻¹(tan x)</strong></p><p>The function tan⁻¹ has range (-π/2, π/2). For any value of tan x, the inverse tan function returns a value in this interval. Therefore:</p><ul><li>When x ∈ (-π/2, π/2): tan⁻¹(tan x) = x</li><li>When x ∉ (-π/2, π/2): tan⁻¹(tan x) ≠ x (it returns an equivalent angle in (-π/2, π/2))</li></ul><p>Thus, tan⁻¹(tan x) is NOT simply x for all real x. Its range is (-π/2, π/2).</p><p><strong>Step 2: Analyze y = cos⁻¹(cos x)</strong></p><p>The function cos⁻¹ has range [0, π]. For any value of cos x, the inverse cosine function returns a value in this interval. Therefore:</p><ul><li>When x ∈ [0, π]: cos⁻¹(cos x) = x</li><li>When x ∉ [0, π]: cos⁻¹(cos x) ≠ x (it returns an equivalent angle in [0, π])</li></ul><p>Thus, cos⁻¹(cos x) is NOT simply x for all real x. Its range is [0, π].</p><p><strong>Step 3: Compare the two functions</strong></p><p>Since tan⁻¹(tan x) and cos⁻¹(cos x) have different ranges ((-π/2, π/2) vs [0, π]), they cannot be the same function. Their graphs differ over various intervals of x. <strong>Statement I is TRUE.</strong></p><p><strong>Step 4: Verify Statement II</strong></p><p>From our analysis above:</p><ul><li>Range of tan⁻¹(tan x) is (-π/2, π/2) ✓</li><li>Range of cos⁻¹(cos x) is [0, π] ✓</li></ul><p><strong>Statement II is TRUE.</strong></p><p><strong>Step 5: Check if Statement II explains Statement I</strong></p><p>Statement II directly identifies the reason why the two functions are different: they have different ranges. The different ranges are the fundamental cause that makes the functions unequal. This is a complete and correct causal explanation.</p><p><strong>Statement II IS a correct explanation for Statement I.</strong></p><p><strong>∴ Answer: a</strong></p>
Correct Answer: a