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[tex]\sf cos(\beta)=\dfrac{1}{4}\implies sec(\beta)=4\\\sf sec^2(\beta)= 1+tg^2(\beta)\\\sf 4^2=1+tg^2(\beta)\\\sf tg^2(\beta)+1=16\\\sf tg^2(\beta)=16-1\\\sf tg^2(\beta)=15\\\sf tg(\beta)=-\sqrt{15}\\\sf cotg(\beta)=\dfrac{1}{tg(\beta)}\\\sf cotg(\beta)=\dfrac{1}{-\sqrt{15}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf cotg(\beta)=-\dfrac{\sqrt{15}}{15}}}}}\blue{\checkmark}[/tex]
