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(A) Given that $\sin \alpha$ and $\cos \alpha$ are the roots of the quadratic equation $ax^2 + bx + c = 0$.From the properties of roots,the sum of roo

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$a^2 - b^2 + 2ac = 0$

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(A) Given that $\sin \alpha$ and $\cos \alpha$ are the roots of the quadratic equation $ax^2 + bx + c = 0$.From the properties of roots,the sum of roots is $\sin \alpha + \cos \alpha = -\frac{b}{a}$ and the product of roots is $\sin \alpha \cos \alpha = \frac{c}{a}$.We know the identity $\sin^2 \alpha + \cos^2 \alpha = 1$.This can be written as $(\sin \alpha + \cos \alpha)^2 - 2\sin \alpha \cos \alpha = 1$.Substituting the values of the sum and product of roots:$(-\frac{b}{a})^2 - 2(\frac{c}{a}) = 1$$\frac{b^2}{a^2} - \frac{2c}{a} = 1$Multiplying both sides by $a^2$:$b^2 - 2ac = a^2$Rearranging the terms,we get $a^2 - b^2 + 2ac = 0$.

If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $ax^2 + bx + c = 0$,then:

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