Multiple choice

Let $\theta$ be an acute angle such that $\displaystyle \sec ^{2}\theta+\tan ^{2}\theta=2 $ The value of $\displaystyle \left ( cosec^{2}\theta+\cot ^{2}\theta \right ) $ is

  1. 9

  2. 5

  3. 4

  4. 2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given sec^2(theta) + tan^2(theta) = 2. Since sec^2 = 1 + tan^2, we have 1 + 2tan^2(theta) = 2, so tan^2(theta) = 1/2. Then sec^2(theta) = 3/2. Cot^2(theta) = 1/tan^2 = 2. Cosec^2(theta) = 1 + cot^2 = 3. Sum = 3 + 2 = 5.

AI explanation

Use the Pythagorean identity for secant and tangent, which states sec squared theta minus tan squared theta equals 1. Adding this identity to the given equation sec squared theta plus tan squared theta equals 2 results in 2 sec squared theta equals 3, meaning sec squared theta equals 1.5 and tan squared theta equals 0.5. We need to evaluate cosec squared theta plus cot squared theta, which can be rewritten using reciprocal identities as 1 plus cot squared theta plus cot squared theta, equaling 1 plus 2 times cot squared theta. Substituting cot squared theta as 1 divided by tan squared theta gives 1 plus 2 times 1 divided by 0.5, resulting in a value of 5.

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