Multiple choice

cosec2θ and sec2θ can be the roots of which of the following quadratic equations?

  1. x2 - x + 1 = 0

  2. x2 - 2x + 2 = 0

  3. x2 - 3x + 3 = 0

  4. x2 - 4x + 4 = 0
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D Correct answer
AI explanation

Using the trigonometric identity cot^2(theta) + tan^2(theta) + 2(cot theta)(tan theta) = (cot theta + tan theta)^2, we can write cosec^2(theta) + sec^2(theta) as (sin^2 theta + cos^2 theta)/(sin^2 theta cos^2 theta), which equals 1/(sin^2 theta cos^2 theta). Substituting sin^2 theta cos^2 theta = (1/4)sin^2(2theta) gives the sum as 4/sin^2(2theta), and their product is cosec^2(theta) sec^2(theta) = 1/(sin^2 theta cos^2 theta) = 4/sin^2(2theta). Since the sum and product are both 4/sin^2(2theta), the required quadratic equation is formed using the sum and product of roots formula x^2 - (sum)x + (product) = 0, yielding x^2 - (4)x + (4) = 0.

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