Step |
Hyp |
Ref |
Expression |
1 |
|
elioore |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → 𝐴 ∈ ℝ ) |
2 |
1
|
recnd |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → 𝐴 ∈ ℂ ) |
3 |
1
|
recoscld |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( cos ‘ 𝐴 ) ∈ ℝ ) |
4 |
|
sincosq1sgn |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( 0 < ( sin ‘ 𝐴 ) ∧ 0 < ( cos ‘ 𝐴 ) ) ) |
5 |
4
|
simprd |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → 0 < ( cos ‘ 𝐴 ) ) |
6 |
3 5
|
elrpd |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( cos ‘ 𝐴 ) ∈ ℝ+ ) |
7 |
6
|
rpne0d |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( cos ‘ 𝐴 ) ≠ 0 ) |
8 |
|
tanval |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) = ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ) |
9 |
2 7 8
|
syl2anc |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( tan ‘ 𝐴 ) = ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ) |
10 |
1
|
resincld |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( sin ‘ 𝐴 ) ∈ ℝ ) |
11 |
4
|
simpld |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → 0 < ( sin ‘ 𝐴 ) ) |
12 |
10 11
|
elrpd |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( sin ‘ 𝐴 ) ∈ ℝ+ ) |
13 |
12 6
|
rpdivcld |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ∈ ℝ+ ) |
14 |
9 13
|
eqeltrd |
⊢ ( 𝐴 ∈ ( 0 (,) ( π / 2 ) ) → ( tan ‘ 𝐴 ) ∈ ℝ+ ) |