Which values represent solutions to Cosine (StartFraction pi Over 4 EndFraction minus x) = StartFraction StartRoot 2 EndRoot Over 2 EndFraction sine x, where x Is an element of [0, 2Pi)?

Left-brace 0, pi right-brace
Left-brace 0, StartFraction 3 pi Over 2 EndFraction right-brace
Left-brace StartFraction pi Over 2 EndFraction, StartFraction 3 pi Over 2 EndFraction right-brace
Left-brace pi, StartFraction 3 pi Over 2 EndFraction

The values that represent the solution to [tex]\mathbf{cos \left(\dfrac{\pi}{4} - x\right)} = \dfrac{\sqrt{2} }{2} \cdot sin \, x[/tex], x ∈ [0, 2·π) are;

[tex]x = \left\{\dfrac{\pi}{2} , \, or \ \dfrac{3 \cdot \pi}{2} \right \}[/tex]

Reason:

The given function is presented as follows;

[tex]cos \left(\dfrac{\pi}{4} - x\right) = \dfrac{\sqrt{2} }{2} \cdot sin \, x[/tex]

Where;

x ∈ [0, 2·π)

We have;

cos(A - B) = cos(A)·cos(B) - sin(A)·sin(B)

[tex]cos \left(\dfrac{\pi}{4} \right) = \dfrac{\sqrt{2} }{2}, \ sin\left(\dfrac{\pi}{4} \right) = \dfrac{\sqrt{2} }{2}[/tex]

Which gives

[tex]cos \left(\dfrac{\pi}{4} - x\right) =\dfrac{\sqrt{2} }{2} \cdot cos \, x + \dfrac{\sqrt{2} }{2} \cdot sin \, x[/tex]

Therefore, we get;

[tex]\dfrac{\sqrt{2} }{2} \cdot cos \, x = \dfrac{\sqrt{2} }{2} \cdot sin \, x - \dfrac{\sqrt{2} }{2} \cdot sin \, x = 0[/tex]

cos x = 0

[tex]cos \left(\dfrac{\pi}{2} \right) = 0[/tex]

Therefore;

The possible solutions (x-values) to cos x = 0 are; [tex]0 \leq \dfrac{n \cdot \pi}{2} \leq 2 \cdot \pi[/tex]

Where;

n = 1, 3, 5, ...

x ∈ [0, 2·π)

We get;

[tex]x = \left\{\dfrac{\pi}{2} , \, or \ \dfrac{3 \cdot \pi}{2} \right \}[/tex]

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