Given that (1+\sin t)(1+\cos t)=\frac{5}{4} and

(1-\sin t)(1-\cos t)=\frac{m}{n}-\sqrt{k}

where k, m, and n are positive integers with m and n relatively prime, find k+m+n.

Given that (1+\sin t)(1+\cos t)=\frac{5}{4} and

(1-\sin t)(1-\cos t)=\frac{m}{n}-\sqrt{k}

where k, m, and n are positive integers with m and n relatively prime, find k+m+n.