I. \(\sin^{-1}(-x) = -\sin^{-1} x\)
II. \(\cos^{-1}(-x) = \cos^{-1} x\)
III. \(\tan^{-1}(-x) = \pi - \tan^{-1} x\)
IV. \(\cot^{-1}(-x) = \pi - \cot^{-1} x\)
How many of the above are correct ?
Inverse Circular Functions Properties Analysis
This question asks to identify the correct statements among the given properties of inverse circular functions. We need to verify each statement.
Statement I: \(\sin^{-1}(-x) = -\sin^{-1} x\)
The inverse sine function, \(\sin^{-1}(x)\), is an odd function. This means it satisfies the property \(\sin^{-1}(-x) = -\sin^{-1} x\) for all \(x\) in its domain, which is [-1, 1]. Therefore, statement I is correct.
Statement II: \(\cos^{-1}(-x) = \cos^{-1} x\)
The inverse cosine function, \(\cos^{-1}(x)\), is neither even nor odd. The correct identity involving \(-x\) is \(\cos^{-1}(-x) = \pi - \cos^{-1} x\) for \(x \in [-1, 1]\). The given statement incorrectly equates \(\cos^{-1}(-x)\) to \(\cos^{-1} x\). Therefore, statement II is incorrect.
Statement III: \(\tan^{-1}(-x) = \pi - \tan^{-1} x\)
The inverse tangent function, \(\tan^{-1}(x)\), is an odd function. It satisfies the property \(\tan^{-1}(-x) = -\tan^{-1} x\) for all \(x\) in its domain, which is \(\mathbb{R}\). The given statement is incorrect. Therefore, statement III is incorrect.
Statement IV: \(\cot^{-1}(-x) = \pi - \cot^{-1} x\)
The inverse cotangent function, \(\cot^{-1}(x)\), satisfies the property \(\cot^{-1}(-x) = \pi - \cot^{-1} x\) for all \(x\) in its domain, which is \(\mathbb{R}\). This property is similar to that of \(\cos^{-1}(x)\). Therefore, statement IV is correct.
Summary of Correct Statements
- Statement I is correct.
- Statement IV is correct.
Out of the four given statements, exactly two statements (I and IV) are correct.
Thus, the number of correct statements is Two.
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