If `[sin^-1 (cos^-1(sin^-1 (tan^-1 x)))]=1`, where `[*]` denotes the greatest integer function, then `x in`

Options

A. `[tan sin cos 1,tan sin cos sin 1]`
B. `(tan sin cos 1,tan sin cos sin1)`
C. `[-1,1]`
D. `[sin cos tan 1,sin cos sin tan 1]`

Answer

To solve the equation \([ \sin^{-1}(\cos^{-1}(\sin^{-1}(\tan^{-1} x))) ] = 1\), where \([ \cdot ]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([a]\) gives the largest integer less than or equal to \(a\). Therefore, the equation \([ \sin^{-1}(\cos^{-1}(\sin^{-1}(\tan^{-1} x))) ] = 1\) implies:
\[ 1 \leq \sin^{-1}(\cos^{-1}(\sin^{-1}(\tan^{-1} x))) < 2 \]

### Step 2: Analyzing the Range of \(\sin^{-1}\)
The range of \(\sin^{-1}(y)\) is \([- \frac{\pi}{2}, \frac{\pi}{2}]\) or approximately \([-1.57, 1.57]\). Therefore, the maximum value of \(\sin^{-1}(\cos^{-1}(\sin^{-1}(\tan^{-1} x)))\) can be at most \(\frac{\pi}{2}\), which is approximately \(1.57\). Since \(1 < 2\) holds true, we can focus on the lower bound:
\[ \sin^{-1}(\cos^{-1}(\sin^{-1}(\tan^{-1} x))) \geq 1 \]

### Step 3: Solving the Inequality
To solve the inequality \(\sin^{-1}(\cos^{-1}(\sin^{-1}(\tan^{-1} x))) \geq 1\), we take the sine of both sides:
\[ \cos^{-1}(\sin^{-1}(\tan^{-1} x)) \geq \sin(1) \]
Since \(\sin(1) \approx 0.8415\), we can rewrite this as:
\[ \cos^{-1}(\sin^{-1}(\tan^{-1} x)) \geq 0.8415 \]

### Step 4: Analyzing \(\cos^{-1}\)
The function \(\cos^{-1}(y)\) is decreasing, so we can set:
\[ \sin^{-1}(\tan^{-1} x) \leq \cos(0.8415) \]
Calculating \(\cos(0.8415) \approx 0.5403\), we have:
\[ \sin^{-1}(\tan^{-1} x) \leq 0.5403 \]

### Step 5: Solving for \(\tan^{-1} x\)
Taking the sine of both sides gives:
\[ \tan^{-1} x \leq \sin(0.5403) \approx 0.5150 \]
Thus, we have:
\[ x \leq \tan(0.5150) \approx 0.5463 \]

### Step 6: Finding the Interval for \(x\)
Since \(\tan^{-1} x\) is defined for all real \(x\), we conclude:
\[ x \in (-\infty, 0.5463] \]

### Final Answer
Thus, the solution for \(x\) is:
\[ x \in (-\infty, \tan(0.5403)] \]