Question

How do you solve `- cos(x) -tan(x)= -1`?

Answer 1

Solve f(x) = - cos x - tan x = - 1

Explanation:

Call `tan (x/2) = t`. Use the formulas: `cos x = (1 - t^2)/(1 + t^2)` and
`tan x = (2t)/(1 - t^2).`
cos x + tan x = 1
`(1 - t^2)/(1 + t^2) + (2t)/(1 - t^2) = 1`
`(1 - t^2)^2 + 2t(1 + t^2) = 1 - t^4`
`(1 + t^4 - 2t^2) + 2t + 2t^3 = 1 - t^4`
`t^4 + t^3 - t^2 + t = 0`
`t(t^3 + t^2 - t + 1) = 0`
`t = tan (x/2) = 0` --> `x/2 = 0` and `x/2 = pi` --> `x = 0`
and `x = 2pi`
(t^3 + t^2 - t + 1) = 0
Solve by graphing calculator.