If `x_1, a n dx_2` are the roots of `x^2+(sintheta-1)x-1/(2cos^2theta)=0,` then find the maximum value of `x1 2+x2 2dot`

Explanation

To solve the problem step by step, we need to find the maximum value of \( x_1^2 + x_2^2 \) given that \( x_1 \) and \( x_2 \) are the roots of the quadratic equation:

\[ x^2 + (\sin \theta - 1)x - \frac{1}{2 \cos^2 \theta} = 0 \]

### Step 1: Identify the coefficients
The coefficients of the quadratic equation are:
- \( a = 1 \)
- \( b = \sin \theta - 1 \)
- \( c = -\frac{1}{2 \cos^2 \theta} \)

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( x_1 + x_2 = -\frac{b}{a} = 1 - \sin \theta \)
- The product of the roots \( x_1 x_2 = \frac{c}{a} = -\frac{1}{2 \cos^2 \theta} \)

### Step 3: Express \( x_1^2 + x_2^2 \)
We can express \( x_1^2 + x_2^2 \) in terms of the sum and product of the roots:

\[ x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 \]

Substituting the values from Vieta's formulas:

\[ x_1^2 + x_2^2 = (1 - \sin \theta)^2 - 2\left(-\frac{1}{2 \cos^2 \theta}\right) \]

### Step 4: Simplify the expression
Now, simplify the expression:

\[ x_1^2 + x_2^2 = (1 - \sin \theta)^2 + \frac{1}{\cos^2 \theta} \]

Expanding \( (1 - \sin \theta)^2 \):

\[ = 1 - 2\sin \theta + \sin^2 \theta + \frac{1}{\cos^2 \theta} \]

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can rewrite \( \frac{1}{\cos^2 \theta} \) as \( \sec^2 \theta \):

\[ = 1 - 2\sin \theta + (1 - \cos^2 \theta) + \sec^2 \theta \]

This simplifies to:

\[ = 2 - 2\sin \theta + \sec^2 \theta \]

### Step 5: Find the maximum value
To find the maximum value of \( x_1^2 + x_2^2 \), we need to minimize \( \sin \theta \) because of the negative sign in front of it. The minimum value of \( \sin \theta \) is \(-1\).

Substituting \( \sin \theta = -1 \):

\[ x_1^2 + x_2^2 = 2 - 2(-1) + \sec^2(-1) \]

Calculating \( \sec^2(-1) \):

Since \( \sec^2 \theta = 1 + \tan^2 \theta \), and \( \tan(-1) = -\tan(1) \), we find that \( \sec^2(-1) = \sec^2(1) \).

However, for our purpose, we can directly calculate:

\[ x_1^2 + x_2^2 = 2 + 2 + 4 = 4 \]

Thus, the maximum value of \( x_1^2 + x_2^2 \) is:

\[ \boxed{4} \]