If ` y = log ((cos x)/(1 - sin x)), "then " (dy)/(dx)` is equal to

Options

A. tanx
B. cosec x
C. cos x
D. sec x

Explanation

To find the derivative of the function \( y = \log\left(\frac{\cos x}{1 - \sin x}\right) \), we will use the properties of logarithms and differentiation. Here’s the step-by-step solution:

### Step 1: Rewrite the function using logarithm properties
We can use the property of logarithms that states \( \log\left(\frac{a}{b}\right) = \log a - \log b \). Therefore, we can rewrite \( y \) as:
\[ y = \log(\cos x) - \log(1 - \sin x) \]

### Step 2: Differentiate using the chain rule
Now, we will differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}[\log(\cos x)] - \frac{d}{dx}[\log(1 - \sin x)] \]

### Step 3: Apply the derivative of logarithm
The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \). We will apply this to both terms:
1. For \( \log(\cos x) \):
\[ \frac{d}{dx}[\log(\cos x)] = \frac{1}{\cos x} \cdot (-\sin x) = -\frac{\sin x}{\cos x} \]

2. For \( \log(1 - \sin x) \):
\[ \frac{d}{dx}[\log(1 - \sin x)] = \frac{1}{1 - \sin x} \cdot (-\cos x) = -\frac{\cos x}{1 - \sin x} \]

### Step 4: Combine the derivatives
Now, we can combine the results from Step 3:
\[ \frac{dy}{dx} = -\frac{\sin x}{\cos x} + \frac{\cos x}{1 - \sin x} \]

### Step 5: Simplify the expression
To combine these fractions, we need a common denominator, which is \( \cos x(1 - \sin x) \):
\[ \frac{dy}{dx} = -\frac{\sin x(1 - \sin x)}{\cos x(1 - \sin x)} + \frac{\cos^2 x}{\cos x(1 - \sin x)} \]
\[ = \frac{-\sin x + \sin^2 x + \cos^2 x}{\cos x(1 - \sin x)} \]

### Step 6: Use the Pythagorean identity
Using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ \frac{dy}{dx} = \frac{-\sin x + 1}{\cos x(1 - \sin x)} = \frac{1 - \sin x}{\cos x(1 - \sin x)} \]

### Step 7: Cancel the common terms
The \( 1 - \sin x \) terms cancel out:
\[ \frac{dy}{dx} = \frac{1}{\cos x} \]

### Step 8: Final result
Since \( \frac{1}{\cos x} = \sec x \), we can write the final answer as:
\[ \frac{dy}{dx} = \sec x \]