Step |
Hyp |
Ref |
Expression |
1 |
|
cotval |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) ) |
2 |
1
|
oveq1d |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( cot ` A ) ^ 2 ) = ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) |
3 |
2
|
oveq2d |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( cot ` A ) ^ 2 ) ) = ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) ) |
4 |
|
sincossq |
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 ) |
5 |
4
|
oveq1d |
|- ( A e. CC -> ( ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) / ( ( sin ` A ) ^ 2 ) ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) ) |
6 |
5
|
adantr |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) / ( ( sin ` A ) ^ 2 ) ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) ) |
7 |
|
sincl |
|- ( A e. CC -> ( sin ` A ) e. CC ) |
8 |
7
|
sqcld |
|- ( A e. CC -> ( ( sin ` A ) ^ 2 ) e. CC ) |
9 |
8
|
adantr |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( sin ` A ) ^ 2 ) e. CC ) |
10 |
|
sqne0 |
|- ( ( sin ` A ) e. CC -> ( ( ( sin ` A ) ^ 2 ) =/= 0 <-> ( sin ` A ) =/= 0 ) ) |
11 |
7 10
|
syl |
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) =/= 0 <-> ( sin ` A ) =/= 0 ) ) |
12 |
11
|
biimpar |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( sin ` A ) ^ 2 ) =/= 0 ) |
13 |
9 12
|
dividd |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( sin ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) = 1 ) |
14 |
13
|
oveq1d |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( ( sin ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) + ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) = ( 1 + ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) ) |
15 |
|
coscl |
|- ( A e. CC -> ( cos ` A ) e. CC ) |
16 |
15
|
sqcld |
|- ( A e. CC -> ( ( cos ` A ) ^ 2 ) e. CC ) |
17 |
16
|
adantr |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( cos ` A ) ^ 2 ) e. CC ) |
18 |
9 17 9 12
|
divdird |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) / ( ( sin ` A ) ^ 2 ) ) = ( ( ( ( sin ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) + ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) ) |
19 |
15 7
|
jca |
|- ( A e. CC -> ( ( cos ` A ) e. CC /\ ( sin ` A ) e. CC ) ) |
20 |
|
2nn0 |
|- 2 e. NN0 |
21 |
|
expdiv |
|- ( ( ( cos ` A ) e. CC /\ ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) /\ 2 e. NN0 ) -> ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) = ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) |
22 |
20 21
|
mp3an3 |
|- ( ( ( cos ` A ) e. CC /\ ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) ) -> ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) = ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) |
23 |
22
|
anassrs |
|- ( ( ( ( cos ` A ) e. CC /\ ( sin ` A ) e. CC ) /\ ( sin ` A ) =/= 0 ) -> ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) = ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) |
24 |
19 23
|
sylan |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) = ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) |
25 |
24
|
oveq2d |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) = ( 1 + ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) ) |
26 |
14 18 25
|
3eqtr4rd |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) = ( ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) / ( ( sin ` A ) ^ 2 ) ) ) |
27 |
|
cscval |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( csc ` A ) = ( 1 / ( sin ` A ) ) ) |
28 |
27
|
oveq1d |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( csc ` A ) ^ 2 ) = ( ( 1 / ( sin ` A ) ) ^ 2 ) ) |
29 |
|
ax-1cn |
|- 1 e. CC |
30 |
|
expdiv |
|- ( ( 1 e. CC /\ ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) /\ 2 e. NN0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) |
31 |
29 20 30
|
mp3an13 |
|- ( ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) |
32 |
7 31
|
sylan |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) |
33 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
34 |
33
|
oveq1i |
|- ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) |
35 |
32 34
|
eqtrdi |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) ) |
36 |
28 35
|
eqtrd |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( csc ` A ) ^ 2 ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) ) |
37 |
6 26 36
|
3eqtr4rd |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( csc ` A ) ^ 2 ) = ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) ) |
38 |
3 37
|
eqtr4d |
|- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( cot ` A ) ^ 2 ) ) = ( ( csc ` A ) ^ 2 ) ) |