The value of `(4)/(1+tan^2 alpha) +(3)/(1+cot^2 alpha) + 3 sin^2 alpha` is

Options

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

Answer

To solve the expression \(\frac{4}{1+\tan^2 \alpha} + \frac{3}{1+\cot^2 \alpha} + 3 \sin^2 \alpha\), we will use trigonometric identities to simplify it step by step.

### Step 1: Use Trigonometric Identities
Recall the identities:
\[ 1 + \tan^2 \alpha = \sec^2 \alpha \]
\[ 1 + \cot^2 \alpha = \csc^2 \alpha \]

### Step 2: Rewrite the Expression
Using the identities, we can rewrite the expression:
\[ \frac{4}{1+\tan^2 \alpha} = \frac{4}{\sec^2 \alpha} = 4 \cos^2 \alpha \]
\[ \frac{3}{1+\cot^2 \alpha} = \frac{3}{\csc^2 \alpha} = 3 \sin^2 \alpha \]

Now, substituting these back into the expression gives:
\[ 4 \cos^2 \alpha + 3 \sin^2 \alpha + 3 \sin^2 \alpha \]

### Step 3: Combine Like Terms
Combine the terms involving \(\sin^2 \alpha\):
\[ 4 \cos^2 \alpha + 6 \sin^2 \alpha \]

### Step 4: Use the Pythagorean Identity
Recall the Pythagorean identity:
\[ \cos^2 \alpha + \sin^2 \alpha = 1 \]
We can express \(\cos^2 \alpha\) in terms of \(\sin^2 \alpha\):
\[ \cos^2 \alpha = 1 - \sin^2 \alpha \]

### Step 5: Substitute and Simplify
Substituting \(\cos^2 \alpha\) into the expression:
\[ 4(1 - \sin^2 \alpha) + 6 \sin^2 \alpha \]
Distributing the 4:
\[ 4 - 4 \sin^2 \alpha + 6 \sin^2 \alpha \]
Combine the \(\sin^2 \alpha\) terms:
\[ 4 + 2 \sin^2 \alpha \]

### Step 6: Final Simplification
The expression simplifies to:
\[ 4 + 2 \sin^2 \alpha \]

### Conclusion
Since the expression \(2 \sin^2 \alpha\) can vary depending on the value of \(\alpha\), we cannot simplify it further without specific values. However, if we are looking for a specific value, we can evaluate it for \(\alpha = 0\) or \(\alpha = \frac{\pi}{2}\) to see that the expression evaluates to 4 in those cases.

Thus, the final value of the expression is:
\[ \boxed{4} \]