Question

Find the principal solution: $2+\sqrt{3}secx-4cosx=2\sqrt{3}$

Answer 1

$2+\sqrt{3}\sec x-4\cos x=2\sqrt{3}$

$\Rightarrow 2-4\cos x=2\sqrt{3}-\sqrt{3}\sec x$

$\Rightarrow 2(1-2\cos x)=\sqrt{3}(2-\sec x)$

$\Rightarrow \frac{1-2\cos x}{2-\frac{1}{\cos x}}=\frac{\sqrt{3}}{2}$

$\Rightarrow \frac{\cos x(1-2\cos x)}{2\cos x-1}=\frac{\sqrt{3}}{2}$

$\Rightarrow -cosx=\frac{\sqrt{3}}{2}$

$\Rightarrow cosx=\frac{-\sqrt{3}}{2}$ is in second and third quadrant.

$\therefore x=\pi -\frac{\pi }{3},\pi +\frac{\pi }{3}$

$\therefore x=\frac{3\pi -\pi }{3},\frac{3\pi +\pi }{3}$

$\therefore x=\frac{2\pi }{3},\frac{4\pi }{3}$