What is the exact value of $([sin ((3pi)/2) + sec (-855°)] [cos(-5pi) + tan(1080°)] )/( [cot (2pi)/3 + sec(-330°)] tan (8pi)/3 + csc 945°)$?

Let's start by the radians. Just remember that $360º=2pi$, so whenever is written in function of $pi$ follows this proportion; for example, $pi=180º$ because it's half that measure and so on.

(I'll adopt three decimals, ok?)

$sin((3pi)/2)=sin((3*180º)/2)=sin(270º)=-0.176$

$cos(-5pi)=cos(-5*180º)=cos(-900º)=cos(-900+360º+360º+360º)=cos(180º)=-0.598$

$(cot(2pi))/3=(cot(360º))/3=(-0,296)/3=0.099$

$tan(8pi)/3=tan(1440)/3=2.238/3=0.746$

Now, the degrees:

$sec(-855º)=sec(-855+360+360+360)=sec(225)=2.722$

$tan(1080º)=-0.856$

$sec(-330º)=sec(-330+360)=6.483$

$csc(945º)=1.722$

Now, let's just substitute them!

$((-0.176+2.722)(-0.598+0.856))/((0.099+6.483)0.746+1.722)$

$(2.546*0.258)/(4.910+1.722)=0.657/6.632=0.099$

Just remembering: I consisdered only three decimal digits, so this answer is not precise.