Evaluation of $\cos\theta - \sin\theta$ using the given definitions:
Given $\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$ and $\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2}$:
Subtracting the two expressions gives:
$\cos\theta - \sin\theta = \frac{(e^{i\theta} + e^{-i\theta}) - (e^{i\theta} - e^{-i\theta})}{2} = \frac{2e^{-i\theta}}{2} = e^{-i\theta}$.

Option breakdown:

• $e^{i\theta}$ is the result of adding the two given definitions: $\cos\theta + \sin\theta$.
• $e^{-i\theta}$ is the exact result obtained by subtracting $\sin\theta$ from $\cos\theta$.
• $-e^{i\theta}$ has an incorrect negative sign.
• $-e^{-i\theta}$ corresponds to $\sin\theta - \cos\theta$.

Vieta's formulas for quadratic polynomials:
For a quadratic polynomial $ax^2 + bx + c$:
Sum of roots $\alpha + \beta = -\frac{b}{a}$.
Product of roots $\alpha \beta = \frac{c}{a}$.
Here, $a = 1$, $b = -2$, and $c = -8$:
Sum $= -\frac{-2}{1} = 2$.
Product $= \frac{-8}{1} = -8$.

Option breakdown:

• $2, 8$ has the wrong sign for the product.
• $-2, 8$ has reversed signs for both sum and product.
• $-2, -8$ has the wrong sign for the sum.
• $2, -8$ is the exact sum and product of the roots.