Step |
Hyp |
Ref |
Expression |
1 |
|
cos2t |
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( ( 2 x. ( ( cos ` A ) ^ 2 ) ) - 1 ) ) |
2 |
|
2cn |
|- 2 e. CC |
3 |
|
sincl |
|- ( A e. CC -> ( sin ` A ) e. CC ) |
4 |
3
|
sqcld |
|- ( A e. CC -> ( ( sin ` A ) ^ 2 ) e. CC ) |
5 |
|
coscl |
|- ( A e. CC -> ( cos ` A ) e. CC ) |
6 |
5
|
sqcld |
|- ( A e. CC -> ( ( cos ` A ) ^ 2 ) e. CC ) |
7 |
|
adddi |
|- ( ( 2 e. CC /\ ( ( sin ` A ) ^ 2 ) e. CC /\ ( ( cos ` A ) ^ 2 ) e. CC ) -> ( 2 x. ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) ) = ( ( 2 x. ( ( sin ` A ) ^ 2 ) ) + ( 2 x. ( ( cos ` A ) ^ 2 ) ) ) ) |
8 |
2 4 6 7
|
mp3an2i |
|- ( A e. CC -> ( 2 x. ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) ) = ( ( 2 x. ( ( sin ` A ) ^ 2 ) ) + ( 2 x. ( ( cos ` A ) ^ 2 ) ) ) ) |
9 |
|
sincossq |
|- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 ) |
10 |
9
|
oveq2d |
|- ( A e. CC -> ( 2 x. ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) ) = ( 2 x. 1 ) ) |
11 |
8 10
|
eqtr3d |
|- ( A e. CC -> ( ( 2 x. ( ( sin ` A ) ^ 2 ) ) + ( 2 x. ( ( cos ` A ) ^ 2 ) ) ) = ( 2 x. 1 ) ) |
12 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
13 |
11 12
|
eqtrdi |
|- ( A e. CC -> ( ( 2 x. ( ( sin ` A ) ^ 2 ) ) + ( 2 x. ( ( cos ` A ) ^ 2 ) ) ) = 2 ) |
14 |
|
mulcl |
|- ( ( 2 e. CC /\ ( ( sin ` A ) ^ 2 ) e. CC ) -> ( 2 x. ( ( sin ` A ) ^ 2 ) ) e. CC ) |
15 |
2 4 14
|
sylancr |
|- ( A e. CC -> ( 2 x. ( ( sin ` A ) ^ 2 ) ) e. CC ) |
16 |
|
mulcl |
|- ( ( 2 e. CC /\ ( ( cos ` A ) ^ 2 ) e. CC ) -> ( 2 x. ( ( cos ` A ) ^ 2 ) ) e. CC ) |
17 |
2 6 16
|
sylancr |
|- ( A e. CC -> ( 2 x. ( ( cos ` A ) ^ 2 ) ) e. CC ) |
18 |
|
subadd |
|- ( ( 2 e. CC /\ ( 2 x. ( ( sin ` A ) ^ 2 ) ) e. CC /\ ( 2 x. ( ( cos ` A ) ^ 2 ) ) e. CC ) -> ( ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) = ( 2 x. ( ( cos ` A ) ^ 2 ) ) <-> ( ( 2 x. ( ( sin ` A ) ^ 2 ) ) + ( 2 x. ( ( cos ` A ) ^ 2 ) ) ) = 2 ) ) |
19 |
2 15 17 18
|
mp3an2i |
|- ( A e. CC -> ( ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) = ( 2 x. ( ( cos ` A ) ^ 2 ) ) <-> ( ( 2 x. ( ( sin ` A ) ^ 2 ) ) + ( 2 x. ( ( cos ` A ) ^ 2 ) ) ) = 2 ) ) |
20 |
13 19
|
mpbird |
|- ( A e. CC -> ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) = ( 2 x. ( ( cos ` A ) ^ 2 ) ) ) |
21 |
20
|
oveq1d |
|- ( A e. CC -> ( ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) - 1 ) = ( ( 2 x. ( ( cos ` A ) ^ 2 ) ) - 1 ) ) |
22 |
|
ax-1cn |
|- 1 e. CC |
23 |
|
sub32 |
|- ( ( 2 e. CC /\ ( 2 x. ( ( sin ` A ) ^ 2 ) ) e. CC /\ 1 e. CC ) -> ( ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) - 1 ) = ( ( 2 - 1 ) - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) ) |
24 |
2 22 23
|
mp3an13 |
|- ( ( 2 x. ( ( sin ` A ) ^ 2 ) ) e. CC -> ( ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) - 1 ) = ( ( 2 - 1 ) - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) ) |
25 |
15 24
|
syl |
|- ( A e. CC -> ( ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) - 1 ) = ( ( 2 - 1 ) - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) ) |
26 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
27 |
26
|
oveq1i |
|- ( ( 2 - 1 ) - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) = ( 1 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) |
28 |
25 27
|
eqtrdi |
|- ( A e. CC -> ( ( 2 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) - 1 ) = ( 1 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) ) |
29 |
1 21 28
|
3eqtr2d |
|- ( A e. CC -> ( cos ` ( 2 x. A ) ) = ( 1 - ( 2 x. ( ( sin ` A ) ^ 2 ) ) ) ) |