Question 593635
What we know: 

cos(theta)*sec(theta) = 1                          (I)
{{{cot(theta) = cos(theta)/sin(theta)}}}                           (II)
{{{cos^2(theta) + sin^2(theta) = 1 }}}                     (III)
sec(theta) = 4                                                     (IV)
sin(theta) < 0                           &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;    (V)
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From (I)+(IV) => {{{cos(theta) = 1/4}}}  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;              (I*)
From (I*)+(III) => {{{sin(theta) = -sqrt(15)/4}}} or {{{sin(theta) = sqrt(15)/4}}}   (II*) 
From (II*)+(V) =>{{{sin(theta) = -sqrt(15)/4}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;     (III*)
From (II)+(III*) => {{{cot(theta) = (1/4)*(-4/sqrt(15)) = -sqrt(15)/15 = -0.258}}}
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Final answer:  cot(theta) = -0.258