What is a value of (\sin \theta - \cos \theta) ?
For the next two (02) items that follow : Let 12(\tan \theta + \cot \theta) = 25, where 45^\circ < \theta < 90^\circ.
- A. -\frac{1}{5}
- B. -\frac{2}{5}
- C. \frac{1}{5} ✓
- D. \frac{2}{5}
Correct Answer: C. \frac{1}{5}
Explanation
Let x = \tan \theta. The equation becomes 12(x + \frac{1}{x}) = 25, yielding 12x^2 - 25x + 12 = 0. Roots are \frac{4}{3} and \frac{3}{4}. Since 45^\circ < \theta < 90^\circ, \tan \theta > 1, so \tan \theta = \frac{4}{3}. This gives a right triangle with sides 4, 3, 5. Thus \sin \theta = \frac{4}{5} and \cos \theta = \frac{3}{5}. Their difference is \frac{1}{5}.
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