Well, its Simple.
You should know but I'm repeating.
#sin theta = (opposite)/(hypoten use) #
#cos theta = (adjacent)/(hypoten use)#
#tan theta = (opposite)/(adjacent)#
#csc theta = (hypoten use)/(opposite) = 1/sin theta#
#sec theta = (hypoten use)/(adjacent) = 1/cos theta#
#cot theta = (adjacent)/(opposite)= 1/tan theta#
Now if you evaluate #sin theta/cos theta#, you get,
#sin theta/cos theta = ((opposite)/(hypoten use) * (hypoten use)/(adjacent)) = (opposite)/(adjacent) = tan theta#
Now, evaluate #cos theta/sin theta#
#cos theta/sin theta = ((adjacent)/(hypoten use) * (hypoten use)/(opposite)) = (adjacent)/(opposite) = cot theta#
#therefore tan theta + cos theta = sin theta/cos theta + cos theta/sin theta#
Hence Proved
