Given that \((1+\sin t)(1+\cos t)=5/4\) and $$(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},$$ where \(k, m,\) and \(n_{}\) are positive integers with \(m_{}\) and \(n_{}\) relatively prime, find \(k+m+n.\)

(第十三届AIME1995 第7题)