If sin `(sin^(-1)(3/5)+cos^(-1)x)=1` then find value of x.

Answer

To solve the equation \( \sin\left(\sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}(x)\right) = 1 \), we can follow these steps:

### Step 1: Understand the equation
The equation states that the sine of the sum of two angles is equal to 1. The sine function reaches its maximum value of 1 at \( \frac{\pi}{2} \).

### Step 2: Set up the equation
From the equation, we can deduce:
\[ \sin^{-1}\left(\frac{3}{5}\right) + \cos^{-1}(x) = \frac{\pi}{2} \]

### Step 3: Rearrange the equation
Now, we can rearrange this equation to isolate \( \cos^{-1}(x) \):
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}\left(\frac{3}{5}\right) \]

### Step 4: Use the identity
Using the identity \( \cos^{-1}(a) = \sin^{-1}(\sqrt{1 - a^2}) \), we can express \( \cos^{-1}(x) \) in terms of sine:
\[ x = \cos\left(\frac{\pi}{2} - \sin^{-1}\left(\frac{3}{5}\right)\right) = \sin\left(\sin^{-1}\left(\frac{3}{5}\right)\right) \]

### Step 5: Calculate the value of \( x \)
Since \( \sin\left(\sin^{-1}\left(\frac{3}{5}\right)\right) = \frac{3}{5} \), we find:
\[ x = \frac{3}{5} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{3}{5}} \]

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