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1/sec α + tan α = sec α - tan α

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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.

User Alysa
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