Question

$\frac{\cot \theta +cosec\theta }{\sin \theta }-\frac{\cot \theta -cosec\theta }{\sin \theta }=\frac{1}{2}$

Answer 1

$\frac{cot\theta ++cosec\theta }{sin\theta }-\frac{cot\theta -cosec\theta }{sin\theta }=\frac{1}{2}$

$\frac{\frac{cos\theta }{sin\theta }+\frac{1}{sin\theta }}{sin\theta }-\frac{\frac{cos\theta }{sin\theta }-\frac{1}{sin\theta }}{sin\theta }=\frac{1}{2}$

$\frac{1+cos\theta }{si{n}^2\theta }-\frac{cos\theta -1}{si{n}^2\theta }=\frac{1}{2}$

$\frac{1+cos\theta -cos\theta +1}{si{n}^2\theta }=\frac{1}{2}$

$si{n}^2\theta =2$

$sin\theta =\pm \sqrt{2}$