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26 October, 23:27

Express sin (tan^-1 (u) - tan^-1 (v)) algebraically in terms of u and v

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  1. 26 October, 23:47
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    Let tan^-1 (u) = A

    tan A = u, opposite side, u, adjacent side = 1, hypotenuse = √ (u^2+1)

    sin A = u/√ (u^2+1)

    cos A = 1/√ (u^2+1)

    let tan^-1 (v) = B

    tan B = v

    sin B = v/√ (v^2+1)

    cos B = 1/√ (v^2+1)

    cos [ tan^-1 (u) + tan^-1 (v) ]

    = cos (A + B)

    = cos A cos B - sin A sin B

    = (1/√ (u^2+1)) (1/√ (v^2+1)) - (u/√ (u^2+1)) (v/√ (v^2+1))

    = (1 - uv) / √ (u^2+1)) √ (v^2+1))
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