Multiple choice

Let $\displaystyle \alpha =\frac{\tan ^{2}A-\sin ^{2}A}{\tan ^{2}A.\sin ^{2}A}: : and: : \beta =\frac{\cot ^{2}A-\cos ^{2}A}{\cot ^{2}A.\cos ^{2}A}$ (A is acute angle) are the roots of the quadratic equation whose discriminant is D then the most appropriate choice is

  1. D > 0
  2. $\displaystyle D\geq 0$

  3. D = 0

  4. D < 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Simplify alpha: (tan^2 - sin^2)/(tan^2 * sin^2) = (sin^2/cos^2 - sin^2)/(sin^4/cos^2) = (sin^2(1-cos^2)/cos^2)/(sin^4/cos^2) = sin^2 * sin^2 / sin^4 = 1. Similarly, beta = 1. If roots are 1 and 1, the discriminant D = b^2 - 4ac = (-2)^2 - 4(1)(1) = 0.

AI explanation

Simplify the expression for alpha by converting tangent to sine over cosine and factoring out sine squared A, which yields sine squared A times sec squared A minus 1, all divided by sine squared A times tangent squared A. Canceling terms leaves alpha equal to 1. Applying the same algebraic identity to the beta expression leaves beta equal to 1 as well. Since both roots of the quadratic equation are 1, the discriminant is zero.

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