The Trig Scrolls I: Quadruple Trouble

Geometry Level pending

If

\(\cos{x}+\cos{y}-\cos{(x+y)}=3/2\)

\(\begin{cases} { \sin }^{2 }{z}+{ \sin }^{2 }{t}\ge3/2 \\ \cos{z}\cos{t}=1/4 \end{cases},\)

and \((x,y)=(\pm a+2\pi n , \pm b+2\pi n)\),

and \((z,t)=(\pm c+2\pi m,\) \(\pm d+\pi (2p+m))\),

where \(n,m,p \in {Z}\),

evaluate

\(\frac { \left|a \right| +\left| b \right| +\left| c \right|}{ \left| d \right| } \)

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