Step |
Hyp |
Ref |
Expression |
1 |
|
tancl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) ∈ ℂ ) |
2 |
|
tanval |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) = ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ) |
3 |
2
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( tan ‘ 𝐴 ) ↑ 2 ) = ( ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ↑ 2 ) ) |
4 |
|
sincl |
⊢ ( 𝐴 ∈ ℂ → ( sin ‘ 𝐴 ) ∈ ℂ ) |
5 |
4
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( sin ‘ 𝐴 ) ∈ ℂ ) |
6 |
|
coscl |
⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) ∈ ℂ ) |
8 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) ≠ 0 ) |
9 |
5 7 8
|
sqdivd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ↑ 2 ) = ( ( ( sin ‘ 𝐴 ) ↑ 2 ) / ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) |
10 |
3 9
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( tan ‘ 𝐴 ) ↑ 2 ) = ( ( ( sin ‘ 𝐴 ) ↑ 2 ) / ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) |
11 |
5
|
sqcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
12 |
7
|
sqcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( cos ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
13 |
12
|
negcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → - ( ( cos ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
14 |
11 12
|
subnegd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) − - ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( sin ‘ 𝐴 ) ↑ 2 ) + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) |
15 |
|
sincossq |
⊢ ( 𝐴 ∈ ℂ → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = 1 ) |
16 |
15
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) + ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = 1 ) |
17 |
14 16
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) − - ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = 1 ) |
18 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
19 |
18
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → 1 ≠ 0 ) |
20 |
17 19
|
eqnetrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) − - ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ≠ 0 ) |
21 |
11 13 20
|
subne0ad |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) ↑ 2 ) ≠ - ( ( cos ‘ 𝐴 ) ↑ 2 ) ) |
22 |
12
|
mulm1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( - 1 · ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = - ( ( cos ‘ 𝐴 ) ↑ 2 ) ) |
23 |
21 22
|
neeqtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) ↑ 2 ) ≠ ( - 1 · ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) |
24 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
25 |
24
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → - 1 ∈ ℂ ) |
26 |
|
sqne0 |
⊢ ( ( cos ‘ 𝐴 ) ∈ ℂ → ( ( ( cos ‘ 𝐴 ) ↑ 2 ) ≠ 0 ↔ ( cos ‘ 𝐴 ) ≠ 0 ) ) |
27 |
6 26
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( ( cos ‘ 𝐴 ) ↑ 2 ) ≠ 0 ↔ ( cos ‘ 𝐴 ) ≠ 0 ) ) |
28 |
27
|
biimpar |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( cos ‘ 𝐴 ) ↑ 2 ) ≠ 0 ) |
29 |
11 25 12 28
|
divmul3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( sin ‘ 𝐴 ) ↑ 2 ) / ( ( cos ‘ 𝐴 ) ↑ 2 ) ) = - 1 ↔ ( ( sin ‘ 𝐴 ) ↑ 2 ) = ( - 1 · ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) ) |
30 |
29
|
necon3bid |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( ( sin ‘ 𝐴 ) ↑ 2 ) / ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ≠ - 1 ↔ ( ( sin ‘ 𝐴 ) ↑ 2 ) ≠ ( - 1 · ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ) ) |
31 |
23 30
|
mpbird |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( ( sin ‘ 𝐴 ) ↑ 2 ) / ( ( cos ‘ 𝐴 ) ↑ 2 ) ) ≠ - 1 ) |
32 |
10 31
|
eqnetrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( tan ‘ 𝐴 ) ↑ 2 ) ≠ - 1 ) |
33 |
|
atandm3 |
⊢ ( ( tan ‘ 𝐴 ) ∈ dom arctan ↔ ( ( tan ‘ 𝐴 ) ∈ ℂ ∧ ( ( tan ‘ 𝐴 ) ↑ 2 ) ≠ - 1 ) ) |
34 |
1 32 33
|
sylanbrc |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) ∈ dom arctan ) |