What is the simplefied value of `(7)/(sec^(2)theta)+ (3)/(1+cot^(2)theta)+ 4sin^(2) theta`?

Options

A. 3
B. 4
C. 5
D. 7

Answer

To simplify the expression \(\frac{7}{\sec^2 \theta} + \frac{3}{1 + \cot^2 \theta} + 4 \sin^2 \theta\), we will follow these steps:

### Step 1: Rewrite \(\sec^2 \theta\) and \(\cot^2 \theta\)
Recall the trigonometric identities:
- \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\)
- \(1 + \cot^2 \theta = \csc^2 \theta = \frac{1}{\sin^2 \theta}\)

Using these identities, we can rewrite the terms in the expression.

### Step 2: Substitute the identities into the expression
Substituting the identities into the expression gives us:
\[ \frac{7}{\sec^2 \theta} = 7 \cos^2 \theta \]
\[ \frac{3}{1 + \cot^2 \theta} = 3 \sin^2 \theta \]
So, the expression becomes:
\[ 7 \cos^2 \theta + 3 \sin^2 \theta + 4 \sin^2 \theta \]

### Step 3: Combine like terms
Now, combine the terms involving \(\sin^2 \theta\):
\[ 7 \cos^2 \theta + (3 \sin^2 \theta + 4 \sin^2 \theta) = 7 \cos^2 \theta + 7 \sin^2 \theta \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can replace \(\cos^2 \theta + \sin^2 \theta\) with 1:
\[ 7 (\cos^2 \theta + \sin^2 \theta) = 7 \cdot 1 = 7 \]

### Final Answer
The simplified value of the expression is:
\[ \boxed{7} \]