Question

answer with explanation
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Answer 1

$2 \sin( \alpha ) + 15 \cos( {}^{2} \alpha ) = 7 \\ \\ 2 \sin( \alpha ) + 15 \: - 15 \sin( {}^{2} \alpha ) = 7 \\ \\ 15 \sin( {}^{2} \alpha ) - 2 \sin( \alpha ) - 8 = 0 \\ \\ it \: is \: in \: the \: form \: a {x}^{2} + bx + c = 0 \\ \\ so .....by \: solving \: the \: equation\\ \\ (3sin \alpha + 2)(5sin \alpha - 4) = 0 \\ \\ sin \alpha = \frac{4}{5} \: or \: \frac{ - 2}{3}$

$when \: sin \alpha = \frac{4}{5} \\ \\ opposite \: side \: of \: \alpha = 4 \\ hypotenuse = 5 \\ adjacent \: side \: = \sqrt{ {5}^{2} - {4}^{2} } = 3 \\ \\ so....cot \alpha = \frac{adjacent \: side}{opposite \: side} = \frac{3}{4}$

$when \: sin \alpha = \frac{ - 2}{3} \\ \\ opp \: side = 2 \\ adjacent \: side = \sqrt{3 {}^{2} - 2 {}^{2} } = \sqrt{5} \\ \\ cot \alpha = \frac{ \sqrt{5} }{2}$

which is not in the option , so [A] 3/4 is the answer