If `tan(cos^(-1) x) = sin (cot^(-1).(1)/(2))`, then find the value of x

Explanation

To solve the equation \( \tan(\cos^{-1} x) = \sin(\cot^{-1} \frac{1}{2}) \), we will follow these steps:

### Step 1: Rewrite the right-hand side
We start with the right-hand side of the equation:
\[ \sin(\cot^{-1} \frac{1}{2}) \]
Using the identity \( \sin(\cot^{-1} y) = \frac{y}{\sqrt{1+y^2}} \), we can substitute \( y = \frac{1}{2} \):
\[ \sin(\cot^{-1} \frac{1}{2}) = \frac{\frac{1}{2}}{\sqrt{1+\left(\frac{1}{2}\right)^2}} = \frac{\frac{1}{2}}{\sqrt{1+\frac{1}{4}}} = \frac{\frac{1}{2}}{\sqrt{\frac{5}{4}}} = \frac{\frac{1}{2}}{\frac{\sqrt{5}}{2}} = \frac{1}{\sqrt{5}} \]

### Step 2: Rewrite the left-hand side
Next, we rewrite the left-hand side:
\[ \tan(\cos^{-1} x) \]
Using the identity \( \tan(\cos^{-1} x) = \frac{\sqrt{1-x^2}}{x} \):
\[ \tan(\cos^{-1} x) = \frac{\sqrt{1-x^2}}{x} \]

### Step 3: Set the two sides equal
Now we set the two sides equal to each other:
\[ \frac{\sqrt{1-x^2}}{x} = \frac{1}{\sqrt{5}} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ \sqrt{1-x^2} = \frac{x}{\sqrt{5}} \]

### Step 5: Square both sides
Next, we square both sides to eliminate the square root:
\[ 1 - x^2 = \frac{x^2}{5} \]

### Step 6: Multiply through by 5
To eliminate the fraction, we multiply through by 5:
\[ 5(1 - x^2) = x^2 \]
This simplifies to:
\[ 5 - 5x^2 = x^2 \]

### Step 7: Rearrange the equation
Rearranging gives us:
\[ 5 = 6x^2 \]
Thus,
\[ x^2 = \frac{5}{6} \]

### Step 8: Solve for x
Taking the square root of both sides, we find:
\[ x = \sqrt{\frac{5}{6}} = \frac{\sqrt{5}}{\sqrt{6}} = \frac{\sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{30}}{6} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = \frac{\sqrt{30}}{6} \]

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