Solve 2tan squared x + secx = 1?

Question

0<=x<=360

Answer 1

x = 0, 2.300523983 (=0, 131.8103149 degrees) (if of interest, the 2nd radian measure is very close to one-half of the cube root of pi, differing only in and after the 4th decimal place)

I'll show how in a moment. First, como erred in one of his algebra steps and seems not to have checked his answer at the end. His 2nd line is fine, but when he, presumably, multiplies both sides by (cos x)^2, he enters "cos x" on the right-hand side instead of (cos x)^2. That throws off his further work which should lead to a quadratic giving roots of cos x = 1, -2/3.

I approached it a different way: substituting the trig identity of tan x = {(sec x)^2 - 1)^0.5 (since I had a secant term) to get:

2 * [ {(sec x)^2 - 1)^0.5 ]^2 + sec x = 1 dealing with squaring a square root:
2 * { (sec x)^2 - 1) } + sec x = 1 cleaning up the parentheses:
2*(sec x)^2 - 2 + sec x = 1 manipulating terms to get zero on the right-hand side and putting the left in standard form:
2*(sec x)^2 + sec x - 3 = 0 quadratic, variable is "sec x":
sec x = 1, -3/2 changing to cos x:
1/(cos x) = 1, -3/2 solving each in turn:

1/(cos x) = 1 multiply each side by "cos x":
1 = cos x find the angle:
x = 0 (and of course, 2Π, 4Π, 6Π, etc.)

1/(cos x) = -3/2 multiply each side by "cos x" and by "-3/2":
-2/3 = cos x find the angle:
x = 2.300523983

So, x = 0, 2.300523983

Which checks out as well.

Answer 2

2 tan ² x + 1 / cos x = 1
2 sin ² x / cos ² x + 1 / cos x = 1
2 sin ² x + cos x = cos x
2(1 - cos ² x) = 0
cos ² x = 1
cos x = ± 1
x = 0° , 180° , 360°

Answer 3

i tried...but cant....