To simplify the given equation, we can rewrite tan α as sin α / cos α.
1/sec α + sin α / cos α = sec α - sin α / cos α
Multiplying both sides of the equation by cos α to clear the denominators:
cos α + sin α = sec α - sin α
Next, we can rewrite sec α as 1 / cos α:
cos α + sin α = 1 / cos α - sin α
Adding sin α to both sides:
cos α + 2sin α = 1 / cos α
Multiplying both sides by cos α:
cos^2 α + 2sin α cos α = 1
Since cos^2 α = 1 - sin^2 α, we can substitute this into the equation:
1 - sin^2 α + 2sin α cos α = 1
Rearranging terms:
2sin α cos α + sin^2 α = 0
Factoring out sin α:
sin α(2cos α + sin α) = 0
Thus, sin α = 0 or 2cos α + sin α = 0.
If sin α = 0, then α can be any multiple of π since sin α = 0 for those values of α.
If 2cos α + sin α = 0, we can rearrange terms:
sin α = -2cos α
Squaring both sides:
sin^2 α = 4cos^2 α
Using the trigonometric identity cos^2 α = 1 - sin^2 α, we can substitute this in:
sin^2 α = 4(1 - sin^2 α)
Expanding:
sin^2 α = 4 - 4sin^2 α
Combining like terms:
5sin^2 α = 4
Dividing by 5:
sin^2 α = 4/5
Taking the square root of both sides:
sin α = ± √(4/5)
Considering the values between 0 and 2π, the possible values for α are:
α = 0, π/2, π, 3π/2, 2π
Thus, the solutions for the equation are α = 0, π/2, π, 3π/2, 2π, and any multiple of π.