Question:

The interval in which the function \( f(x) = \tan^{-1}(\sin x + \cos x) \) is an increasing function, is:

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To determine monotonicity of a composite function, check the derivative of the inner function if the outer function is monotonic. Use domain-specific trigonometric inequalities like \( \tan x<1 \) to identify intervals.
Updated On: Jun 5, 2025
  • \( \left(0, \frac{\pi}{2}\right) \)
  • \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)
  • \( \left(-\frac{3\pi}{4}, \frac{\pi}{4} \right) \)
  • \( \left(\frac{\pi}{4}, \frac{\pi}{2} \right) \)
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The Correct Option is C

Solution and Explanation

Step 1: Let \( f(x) = \tan^{-1}(\sin x + \cos x) \).
Since \(\tan^{-1}(x)\) is an increasing function, \(f(x)\) is increasing where \(\sin x + \cos x\) is increasing.
Step 2: Let \( g(x) = \sin x + \cos x \).
Differentiate:
\[ g'(x) = \cos x - \sin x \] Set \( g'(x) > 0 \) for increasing:
\[ \cos x - \sin x > 0 \quad \Rightarrow \quad \tan x < 1 \] Step 3: Solve inequality \( \tan x < 1 \).
\[ \tan x < 1 \Rightarrow x \in \left(-\frac{3\pi}{4}, \frac{\pi}{4} \right) \pmod{\pi} \] So in this interval, \( \sin x + \cos x \) is increasing, and hence \( f(x) \) is increasing.
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