Line in brown followed by fellow (7)
I believe the answer is:
Here is my best explanation:
'line' is the definition.
(tangent is a kind of line)
'brown followed by fellow' is the subsidiary indication.
'brown' becomes 'tan' (tan is a kind of brown).
'followed by' means one lot of letters go next to another.
'fellow' becomes 'gent' ('gent' can be a synonym of 'fellow').
'in' is the link.
(Other definitions for tangent that I've seen before include "Line touching curve at one point", "mathematician's line", "Straight line that touches but does not intersect a curve", "Different line of thought", "Straight line touching a circle", "Straight line that touches but does not intersect a circle", "Function", "mostly irrelevant item", "Particular line" and "What people often go off at!".)