Question:
Let \(g(x)=ax+b\), where \(a<0\) and \(g\) is defined from \([1,3]\) onto \([0,2]\). Then the value of
\[
\cot\left(\cos^{-1}(|\sin x|+|\cos x|)+\sin^{-1}(-|\cos x|-|\sin x|)\right)
\]
is equal to:
Let \(g(x)=ax+b\), where \(a<0\) and \(g\) is defined from \([1,3]\) onto \([0,2]\). Then the value of
\[
\cot\left(\cos^{-1}(|\sin x|+|\cos x|)+\sin^{-1}(-|\cos x|-|\sin x|)\right)
\]
is equal to:
Show Hint
Whenever you see a complicated inverse trigonometric problem, always check the domain limits of the arguments first. Here, noticing that $|\sin x| + |\cos x|$ is almost always greater than 1 means the function can only exist at the single points where the expression equals exactly 1. This collapses the entire problem down to a few basic angle steps!
Updated On: May 25, 2026
- $g(2)+g(3)$
- $g(2)$
- $g(3)$
- $g(1)+g(2)$
Show Solution
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The Correct Option is C
Solution and Explanation
Concept:
We can evaluate this inverse trigonometric expression by analyzing the properties of complementary angles. Recall the fundamental inverse trigonometric identity for matching arguments:
\[
\sin^{-1}y + \cos^{-1}y = \frac{\pi}{2}
\]
We can handle negative arguments inside the inverse sine function using its odd symmetry property: $\sin^{-1}(-y) = -\sin^{-1}y$.
Step 1: Simplify the internal inverse trigonometric expression.
Let the complicated angle expression inside the cotangent function be defined as $\theta$: \[ \theta = \cos^{-1}(|\sin x|+|\cos x|)+\sin^{-1}(-|\cos x|-|\sin x|) \] Let us define a placeholder variable for the recurring absolute value sum: $y = |\sin x| + |\cos x|$. Using the odd function property of the inverse sine layer, pull out the negative sign: \[ \theta = \cos^{-1}(y) + \sin^{-1}(-y) = \cos^{-1}(y) - \sin^{-1}(y) \]
Step 2: Evaluate the specific range value of the variable tracking parameter.
Let us use the standard trigonometric identity to evaluate the boundary range of $y = |\sin x| + |\cos x|$. Squaring the expression: \[ y^2 = (|\sin x| + |\cos x|)^2 = \sin^2 x + \cos^2 x + 2|\sin x\cos x| = 1 + |\sin 2x| \] Since $0 \le |\sin 2x| \le 1$, the squared range is $1 \le y^2 \le 2 \implies 1 \le y \le \sqrt{2}$. The domain of existence for both $\sin^{-1}y$ and $\cos^{-1}y$ requires that the argument sit strictly within the boundary domain $[-1, 1]$. The only single value that satisfies both the expression range and the domain rules is the endpoint: \[ y = 1 \]
Step 3: Calculate the numerical value of the cotangent function.
Substitute the value $y = 1$ back into our angle equation: \[ \theta = \cos^{-1}(1) - \sin^{-1}(1) = 0 - \frac{\pi}{2} = -\frac{\pi}{2} \] Now substitute this angle back into the main cotangent function expression: \[ \text{Value} = \cot\theta = \cot\left(-\frac{\pi}{2}\right) = 0 \]
Step 4: Map the numerical result to the linear function choices.
We are given that $g(x) = ax + b$ is a decreasing linear function ($a < 0$) that maps the domain interval $[1, 3]$ onto the output range interval $[0, 2]$. Since the function is decreasing, it maps the largest input to the smallest output:
• Lower boundary: $g(1) = 2$
• Upper boundary: $g(3) = 0$ Our evaluated trigonometric result is exactly 0. Comparing this with our boundary outputs shows that: \[ \text{Value} = 0 = g(3) \] This matches option (C) perfectly.
Let the complicated angle expression inside the cotangent function be defined as $\theta$: \[ \theta = \cos^{-1}(|\sin x|+|\cos x|)+\sin^{-1}(-|\cos x|-|\sin x|) \] Let us define a placeholder variable for the recurring absolute value sum: $y = |\sin x| + |\cos x|$. Using the odd function property of the inverse sine layer, pull out the negative sign: \[ \theta = \cos^{-1}(y) + \sin^{-1}(-y) = \cos^{-1}(y) - \sin^{-1}(y) \]
Step 2: Evaluate the specific range value of the variable tracking parameter.
Let us use the standard trigonometric identity to evaluate the boundary range of $y = |\sin x| + |\cos x|$. Squaring the expression: \[ y^2 = (|\sin x| + |\cos x|)^2 = \sin^2 x + \cos^2 x + 2|\sin x\cos x| = 1 + |\sin 2x| \] Since $0 \le |\sin 2x| \le 1$, the squared range is $1 \le y^2 \le 2 \implies 1 \le y \le \sqrt{2}$. The domain of existence for both $\sin^{-1}y$ and $\cos^{-1}y$ requires that the argument sit strictly within the boundary domain $[-1, 1]$. The only single value that satisfies both the expression range and the domain rules is the endpoint: \[ y = 1 \]
Step 3: Calculate the numerical value of the cotangent function.
Substitute the value $y = 1$ back into our angle equation: \[ \theta = \cos^{-1}(1) - \sin^{-1}(1) = 0 - \frac{\pi}{2} = -\frac{\pi}{2} \] Now substitute this angle back into the main cotangent function expression: \[ \text{Value} = \cot\theta = \cot\left(-\frac{\pi}{2}\right) = 0 \]
Step 4: Map the numerical result to the linear function choices.
We are given that $g(x) = ax + b$ is a decreasing linear function ($a < 0$) that maps the domain interval $[1, 3]$ onto the output range interval $[0, 2]$. Since the function is decreasing, it maps the largest input to the smallest output:
• Lower boundary: $g(1) = 2$
• Upper boundary: $g(3) = 0$ Our evaluated trigonometric result is exactly 0. Comparing this with our boundary outputs shows that: \[ \text{Value} = 0 = g(3) \] This matches option (C) perfectly.
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