Mahesh Godavarti

Interesting question. I have to admit that it never occurred to me when I was in high-school to ask this question. I could figure out where the co- in cosine, cotangent and cosecant comes from. It's to do with the complementary angle. I.e.

\sin \alpha = \cos \beta where \alpha + \beta = 90 \degree . i.e. \alpha \text{ and } \beta are complementary angles.

I had to look up where Sine, Tangent, Secant and Cosecant came from.

So, I went to and looked up their meaning.

Origin of sine

1585–95; < New Latin, Latin sinus a curve, fold, pocket, translation of Arabic jayb literally, pocket, by folketymology < Sanskrit jiyā, jyā chord of an arc, literally, bowstring

Origin of tangent

1585–90; < Latin tangent- (stem of tangēns, present participle of tangere to touch) in phrase līnea tangēnstouching line; see -ent

Secant is actually pretty interesting!

(originally) a line from the center of a circle through one extremity of an arc to the tangent from the other extremity.
the ratio of the length of this line to that of the radius of the circle; the reciprocal of the cosine of a given angle or arc. Abbreviation: sec