Proceedings of the Edinburgh Mathematical Society

Cover Proceedings of the Edinburgh Mathematical Society
Proceedings of the Edinburgh Mathematical Society
Edinburgh Mathematical Society
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Digitized by Google 135 3. Required the development of sin~^x. By similar treatment of the identity sin"' 03 + sin~^y - sin~'(a; Jl -y^ + y J\ -x^) we get sin~*a; = a, / J^^ 3_i^ 5_.x' 35^ 63aj" 2SW^ \ V "^ 2 3 "*■ T^T "^ "FT '^2^~9 "'■2«n"*"~2^13"^"/ a^ l'3x' l-3'bx' \ -^\^ + Y 24 5 ■^24.67 and a, is otherwise found to be 1.
sin~'a; is thus found to be an odd function of x.
4. The development of sini^, cosi^. This is got from the identity sin~^a3 4- sin'^y = sin~'(£c Ji -y^ + y Jl -x^) or u-\-
...v = sin~^(sin?>*cosv + cositsinv.
It is shown, first, that sinw is an odd function of u , and cosm an even function of u. The series are then assumed, and the co- efficients evaluated as above.
5. The development of tan~'if. This is got from the identity X -\- ti tan~^ + tan"'// = tan~':j — 1 -xy It is first established that tan~^a; is an odd function of x , and then the series is assumed and the coefficients evaluated in the usual way.
6. The development of tani*. Here we have tanw + tan-y = tan(tfc + 1?) - tanifctanvtan(w + v) .


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