Question
If tanθ = sinα−cosαsinα+cosα, then sinα+cosα and
sinα−cosα must be equal to
sinα−cosα must be equal to
Answer: Option A
:
A
We have tanθ = sinα−cosαsinα+cosα
⇒ tanθ = sin(α−π4)cos(α−π4) ⇒ tanθ = tan(α−π4)
⇒ θ = α - π4⇒ α = θ + π4
Hence, sinα+cosα = sin(θ+π4)+cos(θ+π4)
= √2cosθ
And sinα−cosα = sin(θ+π4) - cos(θ+π4)
= 1√2 sinθ + 1√2 cosθ - 1√2 cosθ + 1√2 sinθ
= 2√2 sinθ = √2sinθ=√2sinθ.
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:
A
We have tanθ = sinα−cosαsinα+cosα
⇒ tanθ = sin(α−π4)cos(α−π4) ⇒ tanθ = tan(α−π4)
⇒ θ = α - π4⇒ α = θ + π4
Hence, sinα+cosα = sin(θ+π4)+cos(θ+π4)
= √2cosθ
And sinα−cosα = sin(θ+π4) - cos(θ+π4)
= 1√2 sinθ + 1√2 cosθ - 1√2 cosθ + 1√2 sinθ
= 2√2 sinθ = √2sinθ=√2sinθ.
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