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# sec(arctan u + arccos v)?

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

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- mohanrao dLv 71 decade agoFavorite Answer
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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