Question 7
Given that $$
(1+\sin t)(1+\cos t)=\frac54
$$ and $$
\begin{aligned}(1-\sin t)&(1-\cos t)\\&=\frac mn-\sqrt k,\end{aligned}
$$ where $k,$ $m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$
(1+\sin t)(1+\cos t)=\frac54
$$ and $$
\begin{aligned}(1-\sin t)&(1-\cos t)\\&=\frac mn-\sqrt k,\end{aligned}
$$ where $k,$ $m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$