Find `dy/dx if sinx=cosy`

Answer

To find \(\frac{dy}{dx}\) given that \(\sin x = \cos y\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Start with the given equation
We have:
\[ \sin x = \cos y \]

### Step 2: Differentiate both sides with respect to \(x\)
Using implicit differentiation, we differentiate both sides:
\[ \frac{d}{dx}(\sin x) = \frac{d}{dx}(\cos y) \]

### Step 3: Apply the differentiation rules
The derivative of \(\sin x\) with respect to \(x\) is \(\cos x\). For the right side, we apply the chain rule. The derivative of \(\cos y\) with respect to \(x\) is:
\[ -\sin y \cdot \frac{dy}{dx} \]
So, we have:
\[ \cos x = -\sin y \cdot \frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Now, we need to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{\cos x}{\sin y} \]

### Step 5: Simplify the expression
Using the identity \(\sin y = \cos(\frac{\pi}{2} - y)\), we can express \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = -\frac{\cos x}{\sin y} \]

This is our final expression for \(\frac{dy}{dx}\).

### Final Answer
\[ \frac{dy}{dx} = -\frac{\cos x}{\sin y} \]

---