<Dd> sin 2 ⁡ (x 2) = 1 − cos ⁡ (x) 2 . (\ displaystyle \ sin ^ (2) \ left ((\ frac (x) (2)) \ right) = (\ frac (1 - \ cos (x)) (2)).) </Dd> <P> Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined . </P> <P> Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed . </P> <P> Some of the early and very significant developments of trigonometry were in India . Influential works from the 4th--5th century, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine . Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476--550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya . The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75 ° intervals from 0 ° to 90 °, to an accuracy of 4 decimal places . They used the words jya for sine, kojya for cosine, utkrama - jya for versine, and otkram jya for inverse sine . The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above . </P>

Where did the words sine cosine and tangent originate from