If tan θ = (sin α − cos α)/(sin α + cos α), where θ and α (α ≠ π/4) are acute angles, then what is √2 sin θ equal to?
A.sin α − cos α✓ Correct
B.sin α + cos α
C.cos α − sin α
D.±(sin α − cos α)
Explanation
tan θ = (sin α − cos α)/(sin α + cos α). Dividing numerator and denominator by cos α: tan θ = (tan α − 1)/(tan α + 1) = tan(α − π/4). So θ = α − π/4 (for acute angles). Then sin θ = sin(α − π/4) = sin α·cos(π/4) − cos α·sin(π/4) = (sin α − cos α)/√2. Therefore √2 sin θ = sin α − cos α.
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