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Step-by-step explanation:
derivative of arctan((cos x)/(1 + sin x)) for x; = d dx [tan^ -1 ( cos^ 2 x 2 -sin^ 2 x 2 cos^ 2 x 2 +sin^ 2 x 2 +2 sin x 2 cos x 2 ]]; = d dx \ tan^ - [ 1-tan( x 2 ) 1+tan( x 2 ) ]][= d dx [tan^ -1 tan( x 4 - x 2 )]= - 1/2
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