solve for x: tanx + secx = 2cosx?

Answer 1

tan x + sec x = 2 cos x
<=> sin x + 1 = 2 cos^2 x (multiplying through by cos x, and noting cos x cannot be 0 since else tan x and sec x are undefined)
<=> sin x + 1 = 2 (1 - sin^2 x)
<=> 2 sin^2 x + sin x - 1 = 0
<=> (2 sin x - 1) (sin x + 1) = 0
<=> sin x = 1/2 or sin x = -1
but we reject sin x = -1 since we know cos x ≠ 0.
So the solution is x = π/6 + 2kπ or x = 5π/6 + 2kπ, for any integer k.

Answer 2

tanx = sinx/cosx, and secx = a million/cosx. sinx/cosx + a million/cosx = 2cosx (sinx + a million)/cosx = 2cosx sinx + a million = 2cos²x sinx = 2cos²x - a million sinx = cos2x sinx - cos2x = 0 sinx - a million - 2sin²x = 0 2sin²x + sinx -a million = 0 sinx => x 2x² + x -a million = 0 (2x - a million)(x + a million) sin x = a million/2, -a million x = ?/6, 3?/2, 5?/6 i think of it rather is sweet... do not carry me to it, although.

Answer 3

sinx/cosx + 1/cosx = 2cosx
sinx + 1 = 2cos^2x
sinx + 1 = 2 - 2sin^2x
2sin^2x + sinx - 1 = 0

sinx = (-1 +/- sqrt(1 + 8))/4
sinx = (-1 +/- 3)/4
sinx = 1/2, -1

-1 is ruled out because tanx is undefined when sinx = -1.

x = sin^-1(1/2), which is pi/6 + k*pi, or 5*pi/6 + k*pi.