Step |
Hyp |
Ref |
Expression |
1 |
|
coscl |
|- ( A e. CC -> ( cos ` A ) e. CC ) |
2 |
1
|
sqcld |
|- ( A e. CC -> ( ( cos ` A ) ^ 2 ) e. CC ) |
3 |
|
ax-1cn |
|- 1 e. CC |
4 |
|
subsub3 |
|- ( ( ( ( cos ` A ) ^ 2 ) e. CC /\ 1 e. CC /\ ( ( cos ` A ) ^ 2 ) e. CC ) -> ( ( ( cos ` A ) ^ 2 ) - ( 1 - ( ( cos ` A ) ^ 2 ) ) ) = ( ( ( ( cos ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) - 1 ) ) |
5 |
3 4
|
mp3an2 |
|- ( ( ( ( cos ` A ) ^ 2 ) e. CC /\ ( ( cos ` A ) ^ 2 ) e. CC ) -> ( ( ( cos ` A ) ^ 2 ) - ( 1 - ( ( cos ` A ) ^ 2 ) ) ) = ( ( ( ( cos ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) - 1 ) ) |
6 |
2 2 5
|
syl2anc |
|- ( A e. CC -> ( ( ( cos ` A ) ^ 2 ) - ( 1 - ( ( cos ` A ) ^ 2 ) ) ) = ( ( ( ( cos ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) - 1 ) ) |
7 |
|
cosadd |
|- ( ( A e. CC /\ A e. CC ) -> ( cos ` ( A + A ) ) = ( ( ( cos ` A ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` A ) ) ) ) |
8 |
7
|
anidms |
|- ( A e. CC -> ( cos ` ( A + A ) ) = ( ( ( cos ` A ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` A ) ) ) ) |
9 |
|
2times |
|- ( A e. CC -> ( 2 x. A ) = ( A + A ) ) |
10 |
9
|
fveq2d |
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( cos ` ( A + A ) ) ) |
11 |
1
|
sqvald |
|- ( A e. CC -> ( ( cos ` A ) ^ 2 ) = ( ( cos ` A ) x. ( cos ` A ) ) ) |
12 |
|
sincl |
|- ( A e. CC -> ( sin ` A ) e. CC ) |
13 |
12
|
sqvald |
|- ( A e. CC -> ( ( sin ` A ) ^ 2 ) = ( ( sin ` A ) x. ( sin ` A ) ) ) |
14 |
11 13
|
oveq12d |
|- ( A e. CC -> ( ( ( cos ` A ) ^ 2 ) - ( ( sin ` A ) ^ 2 ) ) = ( ( ( cos ` A ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` A ) ) ) ) |
15 |
8 10 14
|
3eqtr4d |
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( ( ( cos ` A ) ^ 2 ) - ( ( sin ` A ) ^ 2 ) ) ) |
16 |
12
|
sqcld |
|- ( A e. CC -> ( ( sin ` A ) ^ 2 ) e. CC ) |
17 |
16 2
|
addcomd |
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) ) |
18 |
|
sincossq |
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 ) |
19 |
17 18
|
eqtr3d |
|- ( A e. CC -> ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) = 1 ) |
20 |
|
subadd |
|- ( ( 1 e. CC /\ ( ( cos ` A ) ^ 2 ) e. CC /\ ( ( sin ` A ) ^ 2 ) e. CC ) -> ( ( 1 - ( ( cos ` A ) ^ 2 ) ) = ( ( sin ` A ) ^ 2 ) <-> ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) = 1 ) ) |
21 |
3 2 16 20
|
mp3an2i |
|- ( A e. CC -> ( ( 1 - ( ( cos ` A ) ^ 2 ) ) = ( ( sin ` A ) ^ 2 ) <-> ( ( ( cos ` A ) ^ 2 ) + ( ( sin ` A ) ^ 2 ) ) = 1 ) ) |
22 |
19 21
|
mpbird |
|- ( A e. CC -> ( 1 - ( ( cos ` A ) ^ 2 ) ) = ( ( sin ` A ) ^ 2 ) ) |
23 |
22
|
oveq2d |
|- ( A e. CC -> ( ( ( cos ` A ) ^ 2 ) - ( 1 - ( ( cos ` A ) ^ 2 ) ) ) = ( ( ( cos ` A ) ^ 2 ) - ( ( sin ` A ) ^ 2 ) ) ) |
24 |
15 23
|
eqtr4d |
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( ( ( cos ` A ) ^ 2 ) - ( 1 - ( ( cos ` A ) ^ 2 ) ) ) ) |
25 |
2
|
2timesd |
|- ( A e. CC -> ( 2 x. ( ( cos ` A ) ^ 2 ) ) = ( ( ( cos ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) ) |
26 |
25
|
oveq1d |
|- ( A e. CC -> ( ( 2 x. ( ( cos ` A ) ^ 2 ) ) - 1 ) = ( ( ( ( cos ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) - 1 ) ) |
27 |
6 24 26
|
3eqtr4d |
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( ( 2 x. ( ( cos ` A ) ^ 2 ) ) - 1 ) ) |