Find the tan x + sec x = 2cosx x[0,2pi]

Multiply everything by cos(x): 

sin(x) + 1 = 2cos²(x) 

Rewrite cos²(x) as 1 - sin²(x): 

sin(x) + 1 = 2 - 2sin²(x) 

Move everything to one side: 

2sin²(x) + sin(x) - 1 = 0 

Now it becomes a quadratic, factor it out: 

[2sin(x) - 1] [sin(x) + 1] = 0 

Set each factor to equal zero: 

2sin(x) - 1 = 0 
2sin(x) = 1 
sin(x) = 1/2 
x = π/6, 5π/6 

sin(x) + 1 = 0 
sin(x) = -1 
x = 3π/2 (extraneous) 
 
Overall: 

x = π/6, 5π/6

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