sec(arctan u + arccos v)?

Write an equivalent algebraic expression containing u & v ...

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  • 1 decade ago
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    sec [ arctan(u) + arccos(v)]

    let arctan(u) = A

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

    sin A = u / √(u^2 + 1) ------(1) and cos A = 1 / √(u^2 + 1)----------(2)

    let arccos(v) = B

    cos B = v -------------(3) ==> sin B = √(1 - cos^2(B)) = √(1 - v^2)---------------------(4)

    now sec [ arctan(u) + arccos(v)] = sec [ A + B ]

    = 1 / cos(A + B) = 1 / [cos A cos B - sin A sin B ]

    substitute values from eqns 1, 2, 3 and 4

    1 / [( 1 / √(u^2 + 1) )* v - ( u /√(u^2 + 1) )* √(1 - v^2) ]

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

    = √(u^2 + 1) / ( v - u√(1 - v^2)

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