Metamath Proof Explorer

Theorem rectan

Description: The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014)

Ref Expression
Assertion rectan 
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 /\ ( cos ` A ) =/= 0 ) -> ( cot ` A ) = ( 1 / ( tan ` A ) ) )

Proof

Step Hyp Ref Expression
1 coscl 
 |-  ( A e. CC -> ( cos ` A ) e. CC )
2 sincl 
 |-  ( A e. CC -> ( sin ` A ) e. CC )
3 recdiv 
 |-  ( ( ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) /\ ( ( cos ` A ) e. CC /\ ( cos ` A ) =/= 0 ) ) -> ( 1 / ( ( sin ` A ) / ( cos ` A ) ) ) = ( ( cos ` A ) / ( sin ` A ) ) )
4 2 3 sylanl1 
 |-  ( ( ( A e. CC /\ ( sin ` A ) =/= 0 ) /\ ( ( cos ` A ) e. CC /\ ( cos ` A ) =/= 0 ) ) -> ( 1 / ( ( sin ` A ) / ( cos ` A ) ) ) = ( ( cos ` A ) / ( sin ` A ) ) )
5 1 4 sylanr1 
 |-  ( ( ( A e. CC /\ ( sin ` A ) =/= 0 ) /\ ( A e. CC /\ ( cos ` A ) =/= 0 ) ) -> ( 1 / ( ( sin ` A ) / ( cos ` A ) ) ) = ( ( cos ` A ) / ( sin ` A ) ) )
6 5 3impdi 
 |-  ( ( A e. CC /\ ( sin ` A ) =/= 0 /\ ( cos ` A ) =/= 0 ) -> ( 1 / ( ( sin ` A ) / ( cos ` A ) ) ) = ( ( cos ` A ) / ( sin ` A ) ) )
7 tanval 
 |-  ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
8 7 3adant2 
 |-  ( ( A e. CC /\ ( sin ` A ) =/= 0 /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
9 8 oveq2d 
 |-  ( ( A e. CC /\ ( sin ` A ) =/= 0 /\ ( cos ` A ) =/= 0 ) -> ( 1 / ( tan ` A ) ) = ( 1 / ( ( sin ` A ) / ( cos ` A ) ) ) )
10 cotval 
 |-  ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) )
11 10 3adant3 
 |-  ( ( A e. CC /\ ( sin ` A ) =/= 0 /\ ( cos ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) )
12 6 9 11 3eqtr4rd 
 |-  ( ( A e. CC /\ ( sin ` A ) =/= 0 /\ ( cos ` A ) =/= 0 ) -> ( cot ` A ) = ( 1 / ( tan ` A ) ) )