Multiple choice

If $x<\theta <\cfrac { \pi }{ 2 } $ & $sin\theta +cos\theta +tan\theta +cot\theta +cec\theta +cosec\theta =7$ then $sin2\theta $ is a root of the equation

  1. ${ x }^{ 2 }-44x+36=0$
  2. ${ x }^{ 2 }-44x-36=0$
  3. ${ x }^{ 2 }+44x+36=0$

  4. None of these.
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A Correct answer
Explanation

Interpreting the garbled terms as sec(theta) and 0 < theta < pi/2, let t = sin(theta) + cos(theta). The given sum leads to t^2 - 8t + 9 = 0, and since sin(2theta) = t^2 - 1, eliminating t gives x^2 - 44x + 36 = 0.

AI explanation

Let y equal sin of theta plus cos of theta. Using the trigonometric identity cosec of theta plus cot of theta equals y divided by (y minus 1) and sec of theta plus tan of theta equals y divided by (y minus 1), the given equation becomes y plus twice y divided by (y minus 1) equals 7. Multiplying by (y minus 1) gives y squared minus 8y plus 7 equals 0, which factors to (y minus 1)(y minus 7) equals 0; since theta is between 0 and pi over 2, y must be 7. Substituting y equals 7 into the identity sine of two times theta equals y squared minus 1 yields 48. Therefore, sine of two times theta is 48, meaning it is a root of the equation x squared minus 44x plus 36 equals 0.

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