Maths · Inverse trigonometrical functions and their properties
Range of =\tan ^{-1}\left[\frac{2}{\pi}\left(2 \tan ^{-1} x-\right.\right. \) \r
Range of \( f(x)=\tan ^{-1}\left[\frac{2}{\pi}\left(2 \tan ^{-1} x-\right.\right. \) \( \left.\left.\sin ^{-1} x+\cot ^{-1} x-\cos ^{-1} x\right)\right] \) contains
- A. Only one integer
- B. More than 2 integers
- C. only two integers
- D. No integer
Step-by-step solution
Simplify f(x) using identities: arcsin x + arccos x = π/2 and arctan x + arccot x = π/2, leading to f(x) = arctan((2/π) arctan x) for x in [-1,1]. Let t = arctan x ∈ [-π/4, π/4], then u = (2/π)t ∈ [-1/2, 1/2], and f = arctan u ∈ [-arctan(1/2), arctan(1/2)]. Since 0 is in this interval and arctan(1/2) < 1, the only integer in the range is 0.
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