The value of `sec theta((1+ sin theta)/(cos theta) + (cos theta)/(1+sin theta))- 2 tan^2 theta` is

Options

A. 4
B. 1
C. 2
D. 0

Answer

To solve the expression \( \sec \theta \left( \frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} \right) - 2 \tan^2 \theta \), we will break it down step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \sec \theta \left( \frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} \right) - 2 \tan^2 \theta \]
Recall that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \).

### Step 2: Simplify the first part
Let's simplify the term inside the parentheses:
\[ \frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} \]
Finding a common denominator:
\[ = \frac{(1 + \sin \theta)^2 + \cos^2 \theta}{\cos \theta (1 + \sin \theta)} \]

### Step 3: Expand the numerator
Now, we expand \( (1 + \sin \theta)^2 \):
\[ (1 + \sin \theta)^2 = 1 + 2\sin \theta + \sin^2 \theta \]
Thus, the numerator becomes:
\[ 1 + 2\sin \theta + \sin^2 \theta + \cos^2 \theta \]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ = 1 + 2\sin \theta + 1 = 2 + 2\sin \theta \]

### Step 4: Substitute back into the expression
Now we have:
\[ \frac{2 + 2\sin \theta}{\cos \theta (1 + \sin \theta)} \]
This simplifies to:
\[ \frac{2(1 + \sin \theta)}{\cos \theta (1 + \sin \theta)} = \frac{2}{\cos \theta} = 2 \sec \theta \]

### Step 5: Substitute into the original expression
Now substituting back into the original expression:
\[ \sec \theta \left( 2 \sec \theta \right) - 2 \tan^2 \theta \]
This simplifies to:
\[ 2 \sec^2 \theta - 2 \tan^2 \theta \]

### Step 6: Use the identity
We know that \( \sec^2 \theta = 1 + \tan^2 \theta \):
\[ 2 \sec^2 \theta = 2(1 + \tan^2 \theta) = 2 + 2 \tan^2 \theta \]
Thus, we have:
\[ 2 + 2 \tan^2 \theta - 2 \tan^2 \theta = 2 \]

### Final Answer
The value of the expression is:
\[ \boxed{2} \]