Question

Given costheta • csctheta - 2costheta = 0, solve for theta (May be multiple answers)

Answer 1

Factor out $\cos\theta$:

$\cos\theta\csc\theta-2\cos\theta=\cos\theta(\csc\theta-2)=\cos\theta\left(\dfrac1{\sin\theta}-2\right)=0$

Then either

$\cos\theta=0\implies\theta=\dfrac\pi2+n\pi$

or

$\dfrac1{\sin\theta}-2=0\implies\sin\theta=\dfrac12\implies\theta=\dfrac\pi6+2n\pi\text{ or }\theta=\dfrac{5\pi}6+2n\pi$

where $n$ is any integer.