{"srcs":[{"typ":"Web","fileName":"","url":"https://us.metamath.org/metamath/set.mm","readInstr":"StopBefore","label":"reccot","resetNestingLevel":true,"allLabels":[]}],"descr":"Prove that the tangent is equal to the reciprocal of the cotangent.","varsText":"","disjText":"","stmts":[{"label":"8","typ":"p","isGoal":false,"cont":"|- ( A e. CC -> ( sin ` A ) e. CC )","jstfText":": sincl"},{"label":"7","typ":"p","isGoal":false,"cont":"|- ( A e. CC -> ( cos ` A ) e. CC )","jstfText":": coscl"},{"label":"1","typ":"p","isGoal":false,"cont":"|- ( ( ( ( cos ` A ) e. CC /\\ ( cos ` A ) =/= 0 ) /\\ ( ( sin ` A ) e. CC /\\ ( sin ` A ) =/= 0 ) ) -> ( 1 / ( ( cos ` A ) / ( sin ` A ) ) ) = ( ( sin ` A ) / ( cos ` A ) ) )","jstfText":": recdiv"},{"label":"9","typ":"p","isGoal":false,"cont":"|- ( ( ( A e. CC /\\ ( cos ` A ) =/= 0 ) /\\ ( ( sin ` A ) e. CC /\\ ( sin ` A ) =/= 0 ) ) -> ( 1 / ( ( cos ` A ) / ( sin ` A ) ) ) = ( ( sin ` A ) / ( cos ` A ) ) )","jstfText":"7 1 : sylanl1"},{"label":"10","typ":"p","isGoal":false,"cont":"|- ( ( ( A e. CC /\\ ( cos ` A ) =/= 0 ) /\\ ( A e. CC /\\ ( sin ` A ) =/= 0 ) ) -> ( 1 / ( ( cos ` A ) / ( sin ` A ) ) ) = ( ( sin ` A ) / ( cos ` A ) ) )","jstfText":"8 9 : sylanr1"},{"label":"11","typ":"p","isGoal":false,"cont":"|- ( ( A e. CC /\\ ( sin ` A ) =/= 0 /\\ ( cos ` A ) =/= 0 ) -> ( 1 / ( ( cos ` A ) / ( sin ` A ) ) ) = ( ( sin ` A ) / ( cos ` A ) ) )","jstfText":"10 : uun2131p1"},{"label":"3","typ":"p","isGoal":false,"cont":"|- ( ( A e. CC /\\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) )","jstfText":": cotval"},{"label":"5","typ":"p","isGoal":false,"cont":"|- ( ( A e. CC /\\ ( sin ` A ) =/= 0 /\\ ( cos ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) )","jstfText":"3 : 3adant3"},{"label":"6","typ":"p","isGoal":false,"cont":"|- ( ( A e. CC /\\ ( sin ` A ) =/= 0 /\\ ( cos ` A ) =/= 0 ) -> ( 1 / ( cot ` A ) ) = ( 1 / ( ( cos ` A ) / ( sin ` A ) ) ) )","jstfText":"5 : oveq2d"},{"label":"2","typ":"p","isGoal":false,"cont":"|- ( ( A e. CC /\\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )","jstfText":": tanval"},{"label":"4","typ":"p","isGoal":false,"cont":"|- ( ( A e. CC /\\ ( sin ` A ) =/= 0 /\\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )","jstfText":"2 : 3adant2"},{"label":"reccot","typ":"p","isGoal":true,"cont":"|- ( ( A e. CC /\\ ( sin ` A ) =/= 0 /\\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( 1 / ( cot ` A ) ) )","jstfText":"11 6 4 : 3eqtr4rd"}]}