If `sin(sinx+cosx)=cos(cosx-sinx),` and largest possible value of `sinxi spi/k ,` then the value of `k` is______.

Explanation

To solve the equation \( \sin(\sin x + \cos x) = \cos(\cos x - \sin x) \) and find the largest possible value of \( \sin x \) in the form \( \frac{\pi}{k} \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \sin(\sin x + \cos x) = \cos(\cos x - \sin x) \]
Using the identity \( \cos A = \sin\left(\frac{\pi}{2} - A\right) \), we can rewrite the right-hand side:
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