Multiple choice

If $\theta$ is an acute angle such that $\sec^2 \theta = 3$. then the value of $\dfrac{\tan^2 \theta - \text{cosec}^2 \theta}{\tan^2 \theta - \text{cosec}^2 \theta}$ is

  1. $\dfrac{4}{7}$
  2. $\dfrac{3}{7}$
  3. $1$
  4. $\dfrac{1}{7}$
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C Correct answer
Explanation

The expression is (tan^2(theta) - cosec^2(theta)) / (tan^2(theta) - cosec^2(theta)). Since the numerator and denominator are identical, the expression simplifies to 1, provided the denominator is not zero. Given sec^2(theta) = 3, tan^2(theta) = sec^2(theta) - 1 = 2, and cosec^2(theta) = 1 + cot^2(theta) = 1 + 1/2 = 1.5, the denominator is 2 - 1.5 = 0.5, which is non-zero.

AI explanation

Because the numerator and the denominator of the fraction are identical expressions, (tan squared theta minus cosec squared theta) divided by (tan squared theta minus cosec squared theta), the value is 1 regardless of the angle. Since the top and bottom match exactly, the calculation does not depend on the given condition that sec squared theta is 3. The result is 1.

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