If `tan alpha ,tan beta ` are th roots of the eqution `x^2+px+q=0 (p != 0)` Then `sin^2(alpha+beta)+p sin(alpha+beta)cos(alpha+beta)+qcos^2(alpha+beta)=`

Options

A. independent of p but dependent on q
B. independent of q but dependent on p
C. independent of both p and q
D. dependent on both p and q

Answer

To solve the problem, we need to find the value of the expression:

\[ \sin^2(\alpha + \beta) + p \sin(\alpha + \beta) \cos(\alpha + \beta) + q \cos^2(\alpha + \beta) \]

given that \(\tan \alpha\) and \(\tan \beta\) are the roots of the equation \(x^2 + px + q = 0\).

### Step 1: Identify the roots and their relationships
From the quadratic equation \(x^2 + px + q = 0\), we know that:

- The sum of the roots (i.e., \(\tan \alpha + \tan \beta\)) is given by:
\[ \tan \alpha + \tan \beta = -p \]

- The product of the roots (i.e., \(\tan \alpha \tan \beta\)) is given by:
\[ \tan \alpha \tan \beta = q \]

### Step 2: Use the tangent addition formula
Using the tangent addition formula, we have:
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]
Substituting the values from Step 1:
\[ \tan(\alpha + \beta) = \frac{-p}{1 - q} \]

### Step 3: Express \(\sin(\alpha + \beta)\) and \(\cos(\alpha + \beta)\)
Using the identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
we can express \(\sin(\alpha + \beta)\) and \(\cos(\alpha + \beta)\) in terms of \(\tan(\alpha + \beta)\):
\[ \sin(\alpha + \beta) = \frac{\tan(\alpha + \beta)}{\sqrt{1 + \tan^2(\alpha + \beta)}} = \frac{-p}{\sqrt{(1 - q)^2 + p^2}} \]
\[ \cos(\alpha + \beta) = \frac{1}{\sqrt{1 + \tan^2(\alpha + \beta)}} = \frac{1 - q}{\sqrt{(1 - q)^2 + p^2}} \]

### Step 4: Substitute into the expression
Now substituting \(\sin(\alpha + \beta)\) and \(\cos(\alpha + \beta)\) into the original expression:
\[ \sin^2(\alpha + \beta) = \left(\frac{-p}{\sqrt{(1 - q)^2 + p^2}}\right)^2 = \frac{p^2}{(1 - q)^2 + p^2} \]
\[ \cos^2(\alpha + \beta) = \left(\frac{1 - q}{\sqrt{(1 - q)^2 + p^2}}\right)^2 = \frac{(1 - q)^2}{(1 - q)^2 + p^2} \]
Now substituting these into the expression:
\[ \sin^2(\alpha + \beta) + p \sin(\alpha + \beta) \cos(\alpha + \beta) + q \cos^2(\alpha + \beta) \]

### Step 5: Simplify the expression
Combining all terms:
\[ \frac{p^2}{(1 - q)^2 + p^2} + p \left(\frac{-p}{\sqrt{(1 - q)^2 + p^2}}\right) \left(\frac{1 - q}{\sqrt{(1 - q)^2 + p^2}}\right) + q \frac{(1 - q)^2}{(1 - q)^2 + p^2} \]
This simplifies to:
\[ \frac{p^2 + q(1 - q)^2 - p^2(1 - q)}{(1 - q)^2 + p^2} \]
After further simplification, we find that the expression reduces to:
\[ q \]

### Final Result
Thus, the value of the expression is:
\[ \boxed{q} \]