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If a sin theta + b …

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If a sin theta + b cos theta = c. Prove that a cos theta - b sin theta = root a² + b² - c²

    • Posted by Renu Bala 4 years, 9 months ago

    CBSE > Class 10 > Mathematics
    • 1 answers

    Gaurav Seth 4 years, 9 months ago

    asinθ + bcosθ = c
    taking square both sides,
    (asinθ + bcosθ)² = c²
    ⇒a²sin²θ + b²cos²θ + 2absinθ.cosθ = c² --------(1)

    Let acosθ - bsinθ = x
    Squaring both sides
    (acosθ - bsinθ)² = x²
    ⇒a²cos²θ + b²sin²θ -2absinθ.cosθ = x² ------(2)

    Add equation (1) and (2),
    a²sin²θ + b²cos²θ +2abinθ.cosθ + a²cos²θ + b²sin²θ -2absinθ.cosθ = c² + x²
    ⇒(a² + b²)cos²θ + (a² +b²)sin²θ = c² + x²
    ⇒(a² + b²)[sin²θ + cos²θ ] = c² + x²
    ⇒(a² + b²) = c² + x² [∵ sin²x + cos²x = 1 ]
    ⇒(a² + b² - c²) = x²
    Take square root both sides,

    Hence, acosθ - bsinθ = 

    1Thank You

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