The range of the function f(x) = (cosec-1 x + cos-1 (1/x)/cosec x) is

Question

The range of the function f(x) = (cosec-1 x + cos-1 (1/x)/cosec x) is (A) (0, π) (B) [-(π/2), (π/2)] (C) [-(π/2), 0] (D) [-(π/2), (π/2))

Tardigrade Admin · Accepted Answer

f(x) = (cosec-1 x + sec-1 x/cosec x) , cos-1(1/x) = sec-1 x f(x) = (π/2) sin x Now, -1 le sin x le 1 ∴ -(π/2) le (π/2) sin x le (π/2) ⇒ - (π/2) le f(x) le (π/2)

Q. The range of the function
$f (x) = \frac{cose c ^{- 1} x + co s ^{- 1} ( 1/ x )}{cosec x}$ is

$(0, π)$

$[- \frac{π}{2}, \frac{π}{2}]$

$[- \frac{π}{2}, 0]$

$[- \frac{π}{2}, \frac{π}{2})$

$f (x) = \frac{cose c ^{- 1} x + se c ^{- 1} x}{cosec x}, co s^{- 1} (1/ x) = se c^{- 1} x$
$f (x) = \frac{π}{2} s in x$
Now, $- 1 \leq s in x \leq 1$
$∴ - \frac{π}{2} \leq \frac{π}{2} s in x \leq \frac{π}{2}$
$\Rightarrow - \frac{π}{2} \leq f (x) \leq \frac{π}{2}$

Q. The range of the function f(x)=cosecxcosec−1x+cos−1(1/x)​ is

(0,π)

[−2π​,2π​]

[−2π​,0]

[−2π​,2π​)