If tan x = 0, the equation is solved. tan x = 0 for x = 0, pi, 2pi (in the interval you have specified), so these are solutions.
If we assume that tan x does not equal 0, we can divided both sides by it which gives:
sqrt(2)cos x = 1, OR
cos x = 1/sqrt(2).
We know that this occurs only when x = pi/4 or x = 7pi/4. tan x is defined for these values so these constitute solutions of the original equation as well.
Therefore, in the interval [0,2pi], the solutions are 0,pi/4,pi,7pi/4,2pi.
You may have meant the interval [0,2pi) in which case you can disregard the last solution above (meaning that you may have meant to say (0 <= x < 2pi).
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If tan x = 0, the equation is solved. tan x = 0 for x = 0, pi, 2pi (in the interval you have specified), so these are solutions.
If we assume that tan x does not equal 0, we can divided both sides by it which gives:
sqrt(2)cos x = 1, OR
cos x = 1/sqrt(2).
We know that this occurs only when x = pi/4 or x = 7pi/4. tan x is defined for these values so these constitute solutions of the original equation as well.
Therefore, in the interval [0,2pi], the solutions are 0,pi/4,pi,7pi/4,2pi.
You may have meant the interval [0,2pi) in which case you can disregard the last solution above (meaning that you may have meant to say (0 <= x < 2pi).