First, we need to find the value of cos(theatre) using the identity: sin^2(theatre) + cos^2(theatre) = 1.
Since sin(theatre) = 5/13, we can square it to get sin^2(theatre):
sin^2(theatre) = (5/13)^2 = 25/169.
Now, we can subtract sin^2(theatre) from 1 to find cos^2(theatre):
cos^2(theatre) = 1 - 25/169 = 144/169.
Taking the square root of both sides, we find:
cos(theatre) = √(144/169) = 12/13.
Now, we can use the tangent identity: tan(90 - theatre) = sin(theatre) / cos(theatre).
tan(90 - theatre) = sin(theatre) / cos(theatre) = (5/13) / (12/13) = 5/12.
Therefore, tan(90 - theatre) simplifies to 5/12.