Help with my Trig hw: Trig equations?

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Remember that sin²(x) + cos²(x) = 1

∴ cos(2x) = cos²(x) - sin²(x) = 1 - 2sin²(x)

cos(2x) - √2 sin(x) = 1

1 - 2sin²(x) - √2 sin(x) = 1

2sin²(x) + √2 sin(x) = 0

sin(x) [ 2sin(x) + √2 ] = 0

sin(x) = 0 and sin(x) = -½√2

First Part where sin(x) = 0

x = πk, k ∈ 𝕫

Second Part where sin(x) = -½√2 that is negative in the 3rd and 4th quadrants

(counter clockwise, preferred )

x = 5π/4 + 2πk, k ∈ 𝕫

x = 7π/4 + 2πk, k ∈ 𝕫

( clockwise )

x = -¾π + 2πk, k ∈ 𝕫

x = -¼π + 2πk, k ∈ 𝕫

Answer:

( preferred )

x = πn, k ∈ 𝕫

x = 5π/4 + 2πn, k ∈ 𝕫

x = 7π/4 + 2πn, k ∈ 𝕫

or

x = πn

x = -¾π + 2πk, k ∈ 𝕫

x = -¼π + 2πk, k ∈ 𝕫

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