Multiple choice

cosec 18° is a root of the equation:

  1. x2 - 2x - 4 = 0

  2. x2 - 2x + 4 = 0

  3. 4x2 + 2x - 1 = 0

  4. x2 + 2x - 4 = 0
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A Correct answer
Explanation

sin 18 = (sqrt(5)-1)/4. cosec 18 = 4/(sqrt(5)-1) = sqrt(5)+1. Let x = sqrt(5)+1, then x-1 = sqrt(5), (x-1)^2 = 5, x^2 - 2x + 1 = 5, x^2 - 2x - 4 = 0.

AI explanation

Using the standard trigonometric identity, the value of cosec 18 degrees equals the square root of 4 plus 2 times the square root of 5. Substituting this value into the equation x2 - 2x - 4 gives (4 plus 2 times the square root of 5) minus 2 times the square root of (4 plus 2 times the square root of 5) minus 4. This simplifies to 2 times the square root of 5 minus twice the square root of the same term, which equals zero because the square root of (4 plus 2 times the square root of 5) exactly simplifies to the square root of 5 plus 1. Thus, cosec 18 degrees is a root of x2 - 2x - 4 equals 0.

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