Solve the equation sec^2x - 2 tan x = 4

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The equation `sec^2x - 2*tan x = 4` has to be solved for x.

Using the basic identity `sin^2x + cos^2x = 1` , it is possible to express sec x in terms of tan x.

`sin^2x + cos^2x = 1`

Divide both sides by `cos^2x`

=> `tan^2x + 1 = sec^2x`

`sec^2x - 2*tan x = 4`

=> `1 + tan^2x - 2*tan x = 4`

=> `tan^2x - 2*tan x - 3 = 0`

=> `tan^2x - 3*tan x + tan x - 3 = 0`

=> `tan x(tan x - 3) +1(tan x - 3) = 0`

=> `(tan x + 1)(tan x - 3) = 0`

=> tan x = -1 and tan x = 3

`x = tan^-1(-1) + n*pi` and `x = tan^-1 3 + n*pi`

`x = -pi/4 + n*pi` and `x = tan^-1 3 + n*pi`

The solution of the equation `sec^2x - 2*tan x = 4` is `-pi/4 + n*pi` and `tan^-1 3 + n*pi`

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