Question

If $\tan \theta +\cot \theta =2$, find $\sin \theta$.

Answer 1

We write $tan\theta +cot\theta$ as $\frac{sin\theta }{cos\theta }+\frac{cos\theta }{sin\theta }$, which is equal to

$\frac{si{n}^2\theta +co{s}^2\theta }{sin\theta cos\theta }$

$=\frac{1}{\left(sin2\theta \right)/2}$, where $sin2\theta =2sin\theta cos\theta$

and we have

$\frac{1}{\left(sin2\theta \right)/2}=2$ which implies $sin2\theta =1$ , $\theta =\frac{\pi }{4}$ and we have

$sin\theta =sin{45}^{\circ }=\frac{1}{\sqrt{2}}$