`cosx{(cosx)/(1-sinx)+(1-sinx)/(cosx)}, is `

Options

A. 1
B. 3
C. 2
D. 4

Answer

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

### Step 1: Write the Expression
The given expression is:
\[ \cos x \left( \frac{\cos x}{1 - \sin x} + \frac{1 - \sin x}{\cos x} \right) \]

### Step 2: Find a Common Denominator
To combine the two fractions inside the parentheses, we need a common denominator. The common denominator is \( \cos x (1 - \sin x) \):
\[ \frac{\cos^2 x}{1 - \sin x} + \frac{(1 - \sin x)^2}{\cos x} \]

### Step 3: Combine the Fractions
Now, we can combine the fractions:
\[ \frac{\cos^2 x (1 - \sin x) + (1 - \sin x)^2 \cos x}{\cos x (1 - \sin x)} \]

### Step 4: Simplify the Numerator
Expanding the numerator:
\[ \cos^2 x (1 - \sin x) + (1 - 2\sin x + \sin^2 x) \cos x \]
This simplifies to:
\[ \cos^2 x - \cos^2 x \sin x + \cos x - 2\sin x \cos x + \sin^2 x \cos x \]

### Step 5: Group Like Terms
Now, we can group the terms:
\[ (\cos^2 x + \sin^2 x) \cos x + \cos x - 2\sin x \cos x - \cos^2 x \sin x \]

### Step 6: Use the Pythagorean Identity
Using the identity \( \cos^2 x + \sin^2 x = 1 \):
\[ \cos x + \cos x - 2\sin x \cos x - \cos^2 x \sin x \]
This simplifies to:
\[ 2\cos x - 2\sin x \cos x - \cos^2 x \sin x \]

### Step 7: Factor Out Common Terms
Factoring out \( \cos x \):
\[ \cos x \left( 2 - 2\sin x - \cos x \sin x \right) \]

### Step 8: Divide by the Denominator
Now, we divide by the denominator \( \cos x (1 - \sin x) \):
\[ \frac{\cos x (2 - 2\sin x - \cos x \sin x)}{\cos x (1 - \sin x)} \]
The \( \cos x \) cancels out:
\[ \frac{2 - 2\sin x - \cos x \sin x}{1 - \sin x} \]

### Step 9: Simplify Further
Now, we can simplify:
\[ 2(1 + \sin x) \div (1 - \sin x) \]
This gives us:
\[ \frac{2(1 + \sin x)}{1 - \sin x} \]

### Step 10: Final Answer
The final simplified expression is:
\[ 2 \]