If `cos^(-1) x-sin^(-1)x=0`, then x is equal to

Options

A. `sqrt2`
B. `1/sqrt2`
C. 1
D. `pm 1/sqrt2`

Answer

To solve the equation \( \cos^{-1} x - \sin^{-1} x = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cos^{-1} x - \sin^{-1} x = 0 \]
This can be rewritten as:
\[ \cos^{-1} x = \sin^{-1} x \]

### Step 2: Use the identity
We know from trigonometric identities that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
Using this identity, we can substitute \( \sin^{-1} x \) in our rewritten equation:
\[ \sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x \]
Since we have \( \cos^{-1} x = \sin^{-1} x \), we can substitute \( \sin^{-1} x \) into the identity:
\[ \sin^{-1} x + \sin^{-1} x = \frac{\pi}{2} \]
This simplifies to:
\[ 2 \sin^{-1} x = \frac{\pi}{2} \]

### Step 3: Solve for \( \sin^{-1} x \)
Now, we divide both sides by 2:
\[ \sin^{-1} x = \frac{\pi}{4} \]

### Step 4: Find \( x \)
To find \( x \), we take the sine of both sides:
\[ x = \sin\left(\frac{\pi}{4}\right) \]
Using the known value of sine at \( \frac{\pi}{4} \):
\[ x = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{\sqrt{2}}{2}} \]

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