How many solutions does -2cosx=1tanx have ?

Question

-2cosx=1tanx

Answer 1

Your equation is equivalent to -2cox x = (sin x)/(cos x)

sin x = -2cos²x

sin x = -2(1-sin²x)

2sin²x - sin x  - 2 = 0

let y = sin x; then 2y² - y - 2 = 0 has discriminant (-1)² - 4(2)(-2) = 1 + 17 and y = (1 ±√17) / 4;

using the + sign preceding square root of 17 yields y =1.28... which cannot be any sine value, so y = (1-√17)/4

which is about -0.780776406. For this solution, there are many values x in either quadrant III or IV; if the problem requested solutions that are positive but < 360 degrees, there would be TWO solutions.