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Question:

Let P = {θ: sinθ - cosθ = √2cosθ} and Q = {θ: sinθ + cosθ = √2sinθ} be two sets. Then:

P⊆Q and Q-P≠∅

P=Q

Q⊆P

P⊆Q

Solution:

sinθ - cosθ = √2cosθ ⇒ sinθ = cosθ + √2cosθ = (√2 + 1)cosθ = (2 + √2)cosθ ⇒ (√2 + 1)sinθ = cosθ ⇒ sinθ + cosθ = √2sinθ ∴ P = Q