Costhetha+sinthetha=k then costhetha-sinthetha=?
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cosθ + sinθ = k
squaring both sides,
cos²θ + sin²θ + 2cosθsinθ = k²
⇒2cosθsinθ = k² -1
(cosθ - sinθ)² = ( cosθ+sinθ)² - 4cosθsinθ
= k² - 2(k²-1)
=2 -k²
∴cosθ - sinθ = √(2-k²)
squaring both sides,
cos²θ + sin²θ + 2cosθsinθ = k²
⇒2cosθsinθ = k² -1
(cosθ - sinθ)² = ( cosθ+sinθ)² - 4cosθsinθ
= k² - 2(k²-1)
=2 -k²
∴cosθ - sinθ = √(2-k²)
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