Question

$3tan\theta +cot\theta =5cosec\theta$. Solve for $\theta$, $0\le \theta \le 90$.

___ ${}^{\circ }$

Answer 1

$\frac{3sin\theta }{cos\theta }+\frac{cos\theta }{sin\theta }=\frac{5}{sin\theta }$

$sin\theta (3si{n}^2\theta +co{s}^2\theta )=5sin\theta cos\theta$

$3(1-co{s}^2\theta )+co{s}^2\theta =5cos\theta$

$2co{s}^2\theta +5cos\theta -3=0$

$2cos\theta [cos\theta +3]-1(cos\theta +3)=0$

$(cos\theta +3)(2cos\theta -1)=0$

$cos\theta =-3$ or $cos\theta =\frac{1}{2}$

$cos\theta =-3$ is not possible as $cos\theta$ cannot be lessthan -1
So, $\theta ={60}^{\circ }$