Step |
Hyp |
Ref |
Expression |
1 |
|
df-tan |
⊢ tan = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) ) |
2 |
|
cnvimass |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ dom cos |
3 |
|
cosf |
⊢ cos : ℂ ⟶ ℂ |
4 |
3
|
fdmi |
⊢ dom cos = ℂ |
5 |
2 4
|
sseqtri |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ℂ |
6 |
5
|
sseli |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → 𝑥 ∈ ℂ ) |
7 |
6
|
sincld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
8 |
6
|
coscld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
9 |
|
ffn |
⊢ ( cos : ℂ ⟶ ℂ → cos Fn ℂ ) |
10 |
|
elpreima |
⊢ ( cos Fn ℂ → ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) ) ) |
11 |
3 9 10
|
mp2b |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) ) |
12 |
|
eldifsni |
⊢ ( ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) → ( cos ‘ 𝑥 ) ≠ 0 ) |
13 |
12
|
adantl |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ≠ 0 ) |
14 |
11 13
|
sylbi |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ≠ 0 ) |
15 |
7 8 14
|
divrecd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) = ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) |
16 |
15
|
mpteq2ia |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) |
17 |
1 16
|
eqtri |
⊢ tan = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) |
18 |
17
|
oveq2i |
⊢ ( ℂ D tan ) = ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) ) |
19 |
|
cnelprrecn |
⊢ ℂ ∈ { ℝ , ℂ } |
20 |
19
|
a1i |
⊢ ( ⊤ → ℂ ∈ { ℝ , ℂ } ) |
21 |
|
difss |
⊢ ( ℂ ∖ { 0 } ) ⊆ ℂ |
22 |
|
imass2 |
⊢ ( ( ℂ ∖ { 0 } ) ⊆ ℂ → ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ( ◡ cos “ ℂ ) ) |
23 |
21 22
|
ax-mp |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ( ◡ cos “ ℂ ) |
24 |
|
fimacnv |
⊢ ( cos : ℂ ⟶ ℂ → ( ◡ cos “ ℂ ) = ℂ ) |
25 |
3 24
|
ax-mp |
⊢ ( ◡ cos “ ℂ ) = ℂ |
26 |
23 25
|
sseqtri |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ℂ |
27 |
26
|
sseli |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → 𝑥 ∈ ℂ ) |
28 |
27
|
sincld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
29 |
28
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
30 |
8
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
31 |
|
sincl |
⊢ ( 𝑥 ∈ ℂ → ( sin ‘ 𝑥 ) ∈ ℂ ) |
32 |
31
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
33 |
|
coscl |
⊢ ( 𝑥 ∈ ℂ → ( cos ‘ 𝑥 ) ∈ ℂ ) |
34 |
33
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
35 |
|
dvsin |
⊢ ( ℂ D sin ) = cos |
36 |
|
sinf |
⊢ sin : ℂ ⟶ ℂ |
37 |
36
|
a1i |
⊢ ( ⊤ → sin : ℂ ⟶ ℂ ) |
38 |
37
|
feqmptd |
⊢ ( ⊤ → sin = ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) |
39 |
38
|
oveq2d |
⊢ ( ⊤ → ( ℂ D sin ) = ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) ) |
40 |
3
|
a1i |
⊢ ( ⊤ → cos : ℂ ⟶ ℂ ) |
41 |
40
|
feqmptd |
⊢ ( ⊤ → cos = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) |
42 |
35 39 41
|
3eqtr3a |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) |
43 |
26
|
a1i |
⊢ ( ⊤ → ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ℂ ) |
44 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
45 |
44
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
46 |
45
|
toponrestid |
⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) |
47 |
|
dvtanlem |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ∈ ( TopOpen ‘ ℂfld ) |
48 |
47
|
a1i |
⊢ ( ⊤ → ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ∈ ( TopOpen ‘ ℂfld ) ) |
49 |
20 32 34 42 43 46 44 48
|
dvmptres |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( sin ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( cos ‘ 𝑥 ) ) ) |
50 |
8 14
|
reccld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( 1 / ( cos ‘ 𝑥 ) ) ∈ ℂ ) |
51 |
50
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( 1 / ( cos ‘ 𝑥 ) ) ∈ ℂ ) |
52 |
|
ovexd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) ∈ V ) |
53 |
11
|
simprbi |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) |
54 |
53
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) |
55 |
29
|
negcld |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → - ( sin ‘ 𝑥 ) ∈ ℂ ) |
56 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → 𝑦 ∈ ℂ ) |
57 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → 𝑦 ≠ 0 ) |
58 |
56 57
|
reccld |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → ( 1 / 𝑦 ) ∈ ℂ ) |
59 |
58
|
adantl |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 1 / 𝑦 ) ∈ ℂ ) |
60 |
|
negex |
⊢ - ( 1 / ( 𝑦 ↑ 2 ) ) ∈ V |
61 |
60
|
a1i |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → - ( 1 / ( 𝑦 ↑ 2 ) ) ∈ V ) |
62 |
32
|
negcld |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → - ( sin ‘ 𝑥 ) ∈ ℂ ) |
63 |
41
|
oveq2d |
⊢ ( ⊤ → ( ℂ D cos ) = ( ℂ D ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) ) |
64 |
|
dvcos |
⊢ ( ℂ D cos ) = ( 𝑥 ∈ ℂ ↦ - ( sin ‘ 𝑥 ) ) |
65 |
63 64
|
eqtr3di |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ - ( sin ‘ 𝑥 ) ) ) |
66 |
20 34 62 65 43 46 44 48
|
dvmptres |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ - ( sin ‘ 𝑥 ) ) ) |
67 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
68 |
|
dvrec |
⊢ ( 1 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( 1 / 𝑦 ) ) ) = ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ - ( 1 / ( 𝑦 ↑ 2 ) ) ) ) |
69 |
67 68
|
mp1i |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( 1 / 𝑦 ) ) ) = ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ - ( 1 / ( 𝑦 ↑ 2 ) ) ) ) |
70 |
|
oveq2 |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → ( 1 / 𝑦 ) = ( 1 / ( cos ‘ 𝑥 ) ) ) |
71 |
|
oveq1 |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → ( 𝑦 ↑ 2 ) = ( ( cos ‘ 𝑥 ) ↑ 2 ) ) |
72 |
71
|
oveq2d |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → ( 1 / ( 𝑦 ↑ 2 ) ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
73 |
72
|
negeqd |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → - ( 1 / ( 𝑦 ↑ 2 ) ) = - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
74 |
20 20 54 55 59 61 66 69 70 73
|
dvmptco |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( 1 / ( cos ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) ) ) |
75 |
20 29 30 49 51 52 74
|
dvmptmul |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) ) |
76 |
75
|
mptru |
⊢ ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) |
77 |
|
ovex |
⊢ ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) ∈ V |
78 |
77 1
|
dmmpti |
⊢ dom tan = ( ◡ cos “ ( ℂ ∖ { 0 } ) ) |
79 |
78
|
eqcomi |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) = dom tan |
80 |
8
|
sqcld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) ↑ 2 ) ∈ ℂ ) |
81 |
7
|
sqcld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) ↑ 2 ) ∈ ℂ ) |
82 |
|
sqne0 |
⊢ ( ( cos ‘ 𝑥 ) ∈ ℂ → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) ≠ 0 ↔ ( cos ‘ 𝑥 ) ≠ 0 ) ) |
83 |
8 82
|
syl |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) ≠ 0 ↔ ( cos ‘ 𝑥 ) ≠ 0 ) ) |
84 |
14 83
|
mpbird |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) ↑ 2 ) ≠ 0 ) |
85 |
80 81 80 84
|
divdird |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) + ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) ) |
86 |
80 81
|
addcomd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( sin ‘ 𝑥 ) ↑ 2 ) + ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
87 |
|
sincossq |
⊢ ( 𝑥 ∈ ℂ → ( ( ( sin ‘ 𝑥 ) ↑ 2 ) + ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
88 |
6 87
|
syl |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( sin ‘ 𝑥 ) ↑ 2 ) + ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
89 |
86 88
|
eqtrd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
90 |
89
|
oveq1d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
91 |
85 90
|
eqtr3d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) + ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
92 |
8 14
|
recidd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) = 1 ) |
93 |
80 84
|
dividd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
94 |
92 93
|
eqtr4d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) = ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
95 |
7 7 80 84
|
div23d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
96 |
7
|
sqvald |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) ↑ 2 ) = ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) |
97 |
96
|
oveq1d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
98 |
80 84
|
reccld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ∈ ℂ ) |
99 |
98 7
|
mul2negd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) = ( ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
100 |
7 80 84
|
divrec2d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
101 |
99 100
|
eqtr4d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) = ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
102 |
101
|
oveq1d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) = ( ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
103 |
95 97 102
|
3eqtr4rd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) = ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
104 |
94 103
|
oveq12d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) + ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) ) |
105 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
106 |
|
expneg |
⊢ ( ( ( cos ‘ 𝑥 ) ∈ ℂ ∧ 2 ∈ ℕ0 ) → ( ( cos ‘ 𝑥 ) ↑ - 2 ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
107 |
8 105 106
|
sylancl |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) ↑ - 2 ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
108 |
91 104 107
|
3eqtr4d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) |
109 |
108
|
rgen |
⊢ ∀ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) ↑ - 2 ) |
110 |
|
mpteq12 |
⊢ ( ( ( ◡ cos “ ( ℂ ∖ { 0 } ) ) = dom tan ∧ ∀ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) → ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ dom tan ↦ ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) ) |
111 |
79 109 110
|
mp2an |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ dom tan ↦ ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) |
112 |
18 76 111
|
3eqtri |
⊢ ( ℂ D tan ) = ( 𝑥 ∈ dom tan ↦ ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) |