let r be Real; :: thesis: ( 1 <= r & r <= sqrt 2 implies ( sin . (arcsec1 r) = (sqrt ((r ^2 ) - 1)) / r & cos . (arcsec1 r) = 1 / r ) )
set x = arcsec1 r;
assume A1: ( 1 <= r & r <= sqrt 2 ) ; :: thesis: ( sin . (arcsec1 r) = (sqrt ((r ^2 ) - 1)) / r & cos . (arcsec1 r) = 1 / r )
then r in [.1,(sqrt 2).] ;
then A2: arcsec1 r in dom (sec | [.0 ,(PI / 4).]) by Lm13, Th85;
A3: dom (sec | [.0 ,(PI / 4).]) c= dom sec by RELAT_1:89;
A5: r = (cos ^ ) . (arcsec1 r) by A1, Th89
.= 1 / (cos . (arcsec1 r)) by A2, A3, RFUNCT_1:def 8 ;
r > 0 by A1;
then A7: r ^2 > 0 ;
((sin . (arcsec1 r)) ^2 ) + ((cos . (arcsec1 r)) ^2 ) = 1 by SIN_COS:31;
then A8: (sin . (arcsec1 r)) ^2 = 1 - ((cos . (arcsec1 r)) ^2 )
.= 1 - ((1 / r) * (1 / r)) by A5
.= 1 - (1 / (r ^2 )) by XCMPLX_1:103
.= ((r ^2 ) / (r ^2 )) - (1 / (r ^2 )) by A7, XCMPLX_1:60
.= ((r ^2 ) - 1) / (r ^2 ) ;
r ^2 >= 1 ^2 by A1, SQUARE_1:77;
then A9: (r ^2 ) - 1 >= 0 by XREAL_1:50;
A12: 0 in [.0 ,PI .] ;
PI / 4 < PI / 1 by XREAL_1:78;
then PI / 4 in [.0 ,PI .] ;
then [.0 ,(PI / 4).] c= [.0 ,PI .] by A12, XXREAL_2:def 12;
then sin . (arcsec1 r) >= 0 by A2, Lm13, COMPTRIG:24;
then sin . (arcsec1 r) = sqrt (((r ^2 ) - 1) / (r ^2 )) by A8, SQUARE_1:def 4
.= (sqrt ((r ^2 ) - 1)) / (sqrt (r ^2 )) by A7, A9, SQUARE_1:99
.= (sqrt ((r ^2 ) - 1)) / r by A1, SQUARE_1:89 ;
hence ( sin . (arcsec1 r) = (sqrt ((r ^2 ) - 1)) / r & cos . (arcsec1 r) = 1 / r ) by A5; :: thesis: verum