Solve; 3sec^2(x)=2tanx+4

Answer 1

3sec^2(x)= 2tan(x) + 4

3(1+tan^2(x)) -2tan(x) - 4 =0

let z= tan(x)

3z^2 -2z -1= 0

factor

(3z +1)(z-1) =0

set each factor = 0

3z+1 =0

z=-1/3 = tan(x)

x = tan^-1(-1/3)<0 as the angle is in quadrant II or IV

x=about -18.435 degrees + 180n where n= any integer

z-1=0

z=0= tan(x)

x= tan^-1(0)= 0+npi =npi where n= any integer

x= ...-2pi, -pi, 0, pi, 2pi, ...

Answer 2

3(1 + tan²x) = 2tanx + 4; 3tan²x - 2tanx - 1 = 0; (3tanx + 1)(tanx - 1) = 0

tanx = 1; x = π/4 + nπ;

tanx = - 1/3; x = - tan^-1(1/3) + nπ

n = 0, ±1, ±2, ...