Step |
Hyp |
Ref |
Expression |
1 |
|
addcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) |
2 |
1
|
adantr |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( A + B ) e. CC ) |
3 |
|
simpr3 |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` ( A + B ) ) =/= 0 ) |
4 |
|
tanval |
|- ( ( ( A + B ) e. CC /\ ( cos ` ( A + B ) ) =/= 0 ) -> ( tan ` ( A + B ) ) = ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) ) |
5 |
2 3 4
|
syl2anc |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` ( A + B ) ) = ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) ) |
6 |
|
sinadd |
|- ( ( A e. CC /\ B e. CC ) -> ( sin ` ( A + B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) ) |
7 |
6
|
adantr |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( sin ` ( A + B ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) ) |
8 |
|
cosadd |
|- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) |
9 |
8
|
adantr |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) |
10 |
7 9
|
oveq12d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) = ( ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) ) |
11 |
|
simpll |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> A e. CC ) |
12 |
11
|
coscld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` A ) e. CC ) |
13 |
|
simplr |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> B e. CC ) |
14 |
13
|
coscld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` B ) e. CC ) |
15 |
12 14
|
mulcld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC ) |
16 |
|
simpr1 |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` A ) =/= 0 ) |
17 |
11 16
|
tancld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` A ) e. CC ) |
18 |
|
simpr2 |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( cos ` B ) =/= 0 ) |
19 |
13 18
|
tancld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` B ) e. CC ) |
20 |
15 17 19
|
adddid |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) = ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) + ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) ) ) |
21 |
12 14 17
|
mul32d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) = ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( cos ` B ) ) ) |
22 |
|
tanval |
|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) ) |
23 |
11 16 22
|
syl2anc |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) ) |
24 |
23
|
oveq2d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( tan ` A ) ) = ( ( cos ` A ) x. ( ( sin ` A ) / ( cos ` A ) ) ) ) |
25 |
11
|
sincld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( sin ` A ) e. CC ) |
26 |
25 12 16
|
divcan2d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( ( sin ` A ) / ( cos ` A ) ) ) = ( sin ` A ) ) |
27 |
24 26
|
eqtrd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( tan ` A ) ) = ( sin ` A ) ) |
28 |
27
|
oveq1d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( cos ` B ) ) = ( ( sin ` A ) x. ( cos ` B ) ) ) |
29 |
21 28
|
eqtrd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) = ( ( sin ` A ) x. ( cos ` B ) ) ) |
30 |
12 14 19
|
mulassd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) = ( ( cos ` A ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) ) |
31 |
|
tanval |
|- ( ( B e. CC /\ ( cos ` B ) =/= 0 ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) ) |
32 |
13 18 31
|
syl2anc |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` B ) = ( ( sin ` B ) / ( cos ` B ) ) ) |
33 |
32
|
oveq2d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` B ) x. ( tan ` B ) ) = ( ( cos ` B ) x. ( ( sin ` B ) / ( cos ` B ) ) ) ) |
34 |
13
|
sincld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( sin ` B ) e. CC ) |
35 |
34 14 18
|
divcan2d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` B ) x. ( ( sin ` B ) / ( cos ` B ) ) ) = ( sin ` B ) ) |
36 |
33 35
|
eqtrd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` B ) x. ( tan ` B ) ) = ( sin ` B ) ) |
37 |
36
|
oveq2d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) = ( ( cos ` A ) x. ( sin ` B ) ) ) |
38 |
30 37
|
eqtrd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) = ( ( cos ` A ) x. ( sin ` B ) ) ) |
39 |
29 38
|
oveq12d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` A ) ) + ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( tan ` B ) ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) ) |
40 |
20 39
|
eqtrd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) = ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) ) |
41 |
|
1cnd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> 1 e. CC ) |
42 |
17 19
|
mulcld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( tan ` A ) x. ( tan ` B ) ) e. CC ) |
43 |
15 41 42
|
subdid |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. 1 ) - ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) ) ) |
44 |
15
|
mulid1d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. 1 ) = ( ( cos ` A ) x. ( cos ` B ) ) ) |
45 |
12 14 17 19
|
mul4d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) = ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) ) |
46 |
27 36
|
oveq12d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( tan ` A ) ) x. ( ( cos ` B ) x. ( tan ` B ) ) ) = ( ( sin ` A ) x. ( sin ` B ) ) ) |
47 |
45 46
|
eqtrd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) = ( ( sin ` A ) x. ( sin ` B ) ) ) |
48 |
44 47
|
oveq12d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. 1 ) - ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) |
49 |
43 48
|
eqtrd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) |
50 |
40 49
|
oveq12d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) ) = ( ( ( ( sin ` A ) x. ( cos ` B ) ) + ( ( cos ` A ) x. ( sin ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) ) |
51 |
17 19
|
addcld |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( tan ` A ) + ( tan ` B ) ) e. CC ) |
52 |
|
ax-1cn |
|- 1 e. CC |
53 |
|
subcl |
|- ( ( 1 e. CC /\ ( ( tan ` A ) x. ( tan ` B ) ) e. CC ) -> ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) e. CC ) |
54 |
52 42 53
|
sylancr |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) e. CC ) |
55 |
|
tanaddlem |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 ) ) -> ( ( cos ` ( A + B ) ) =/= 0 <-> ( ( tan ` A ) x. ( tan ` B ) ) =/= 1 ) ) |
56 |
55
|
3adantr3 |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` ( A + B ) ) =/= 0 <-> ( ( tan ` A ) x. ( tan ` B ) ) =/= 1 ) ) |
57 |
3 56
|
mpbid |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( tan ` A ) x. ( tan ` B ) ) =/= 1 ) |
58 |
57
|
necomd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> 1 =/= ( ( tan ` A ) x. ( tan ` B ) ) ) |
59 |
|
subeq0 |
|- ( ( 1 e. CC /\ ( ( tan ` A ) x. ( tan ` B ) ) e. CC ) -> ( ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) = 0 <-> 1 = ( ( tan ` A ) x. ( tan ` B ) ) ) ) |
60 |
59
|
necon3bid |
|- ( ( 1 e. CC /\ ( ( tan ` A ) x. ( tan ` B ) ) e. CC ) -> ( ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) =/= 0 <-> 1 =/= ( ( tan ` A ) x. ( tan ` B ) ) ) ) |
61 |
52 42 60
|
sylancr |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) =/= 0 <-> 1 =/= ( ( tan ` A ) x. ( tan ` B ) ) ) ) |
62 |
58 61
|
mpbird |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) =/= 0 ) |
63 |
12 14 16 18
|
mulne0d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( cos ` A ) x. ( cos ` B ) ) =/= 0 ) |
64 |
51 54 15 62 63
|
divcan5d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( ( tan ` A ) + ( tan ` B ) ) ) / ( ( ( cos ` A ) x. ( cos ` B ) ) x. ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) ) = ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) ) |
65 |
10 50 64
|
3eqtr2rd |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) = ( ( sin ` ( A + B ) ) / ( cos ` ( A + B ) ) ) ) |
66 |
5 65
|
eqtr4d |
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) =/= 0 /\ ( cos ` B ) =/= 0 /\ ( cos ` ( A + B ) ) =/= 0 ) ) -> ( tan ` ( A + B ) ) = ( ( ( tan ` A ) + ( tan ` B ) ) / ( 1 - ( ( tan ` A ) x. ( tan ` B ) ) ) ) ) |