If `cos^(-1)x=tan^(-1)x`, then `sin(cos^(-1)x)=`

Options

A. `x`
B. `x^(2)`
C. `1//x`
D. `1//x^(2)`

Answer

To solve the equation \( \cos^{-1} x = \tan^{-1} x \) and find \( \sin(\cos^{-1} x) \), we can follow these steps:

### Step 1: Set up the equation
Let \( y = \cos^{-1} x \). Therefore, we can write:
\[ y = \tan^{-1} x \]

### Step 2: Express \( x \) in terms of \( y \)
From the definition of the inverse cosine function, we have:
\[ \cos y = x \]
From the definition of the inverse tangent function, we have:
\[ \tan y = x \]

### Step 3: Relate \( \tan y \) and \( \cos y \)
Recall that:
\[ \tan y = \frac{\sin y}{\cos y} \]
Substituting \( \cos y = x \) into the equation gives:
\[ \tan y = \frac{\sin y}{x} \]
Since we also know \( \tan y = x \), we can set the two expressions for \( \tan y \) equal to each other:
\[ x = \frac{\sin y}{x} \]

### Step 4: Solve for \( \sin y \)
Multiplying both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 = \sin y \]

### Step 5: Find \( \sin(\cos^{-1} x) \)
Since we have established that \( y = \cos^{-1} x \), we can write:
\[ \sin(\cos^{-1} x) = \sin y = x^2 \]

### Conclusion
Thus, the final answer is:
\[ \sin(\cos^{-1} x) = x^2 \]