Find the derivatives of the following :
`(sin(x+a))/cosx`

Answer

To find the derivative of the function \( y = \frac{\sin(x + a)}{\cos x} \), we can follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function for clarity:
\[ y = \frac{\sin(x + a)}{\cos x} \]

### Step 2: Use the sine addition formula
Using the sine addition formula, we can expand \( \sin(x + a) \):
\[ \sin(x + a) = \sin x \cos a + \cos x \sin a \]
Substituting this back into our function gives:
\[ y = \frac{\sin x \cos a + \cos x \sin a}{\cos x} \]
This simplifies to:
\[ y = \tan x \cos a + \sin a \]

### Step 3: Differentiate the function
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\tan x \cos a) + \frac{d}{dx}(\sin a) \]

### Step 4: Apply the derivative rules
The derivative of \( \tan x \) is \( \sec^2 x \) and since \( \cos a \) is a constant, we have:
\[ \frac{dy}{dx} = \cos a \cdot \sec^2 x + 0 \]
Thus, the derivative simplifies to:
\[ \frac{dy}{dx} = \cos a \sec^2 x \]

### Final Answer
The derivative of the function \( y = \frac{\sin(x + a)}{\cos x} \) is:
\[ \frac{dy}{dx} = \cos a \sec^2 x \]