Which of the following is not the quadratic equation whose roots are cos²θ and sec²θ?

• x² − 6x + 6 = 0
• x² − 7x + 7 = 0
• x² − 4x + 4 = 0
• None of these

To determine which equation is not a quadratic equation with roots cos²θ and sec²θ, we first need to understand the relationship between these roots. The roots of a quadratic equation in the form ax² + bx + c = 0 can be expressed using Vieta's formulas, where the sum of the roots (r1 + r2) is equal to -b/a and the product of the roots (r1 * r2) is equal to c/a.

Calculating the Roots

For the roots cos²θ and sec²θ, we know:

• Sum: cos²θ + sec²θ = cos²θ + 1/cos²θ
• Product: cos²θ * sec²θ = 1

Finding the Sum and Product

Let's calculate the sum:

Let x = cos²θ. Then, the sum becomes:

x + 1/x = (x² + 1)/x.

To find the minimum value of this expression, we can use calculus or the AM-GM inequality, which gives us a minimum of 2 when x = 1 (i.e., cos²θ = 1).

Thus, the sum of the roots is at least 2, and the product is 1.

Analyzing the Given Equations

Now, let's analyze the provided equations:

• x² − 6x + 6 = 0: Sum = 6, Product = 6
• x² − 7x + 7 = 0: Sum = 7, Product = 7
• x² − 4x + 4 = 0: Sum = 4, Product = 4

Identifying the Incorrect Equation

Since the sum of the roots must be at least 2 and the product must be 1, we can see that:

• The first equation has a sum of 6 and product of 6.
• The second equation has a sum of 7 and product of 7.
• The third equation has a sum of 4 and product of 4.

All these equations have sums greater than 2, but we need to find the one that does not fit the requirement of having a product of 1.

Final Answer

The equation x² − 4x + 4 = 0 has a product of 4, which does not match the required product of 1. Therefore, this is the equation that is not a quadratic equation with roots cos²θ and sec²θ.