Theorem cotval

Description: Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014)

	Ref	Expression
Assertion	cotval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( cot ‘ 𝐴 ) = ( ( cos ‘ 𝐴 ) / ( sin ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝑦 = 𝐴 → ( sin ‘ 𝑦 ) = ( sin ‘ 𝐴 ) )
2	1	neeq1d	⊢ ( 𝑦 = 𝐴 → ( ( sin ‘ 𝑦 ) ≠ 0 ↔ ( sin ‘ 𝐴 ) ≠ 0 ) )
3	2	elrab	⊢ ( 𝐴 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↔ ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) )
4		fveq2	⊢ ( 𝑥 = 𝐴 → ( cos ‘ 𝑥 ) = ( cos ‘ 𝐴 ) )
5		fveq2	⊢ ( 𝑥 = 𝐴 → ( sin ‘ 𝑥 ) = ( sin ‘ 𝐴 ) )
6	4 5	oveq12d	⊢ ( 𝑥 = 𝐴 → ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) = ( ( cos ‘ 𝐴 ) / ( sin ‘ 𝐴 ) ) )
7		df-cot	⊢ cot = ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) )
8		ovex	⊢ ( ( cos ‘ 𝐴 ) / ( sin ‘ 𝐴 ) ) ∈ V
9	6 7 8	fvmpt	⊢ ( 𝐴 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( cot ‘ 𝐴 ) = ( ( cos ‘ 𝐴 ) / ( sin ‘ 𝐴 ) ) )
10	3 9	sylbir	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( cot ‘ 𝐴 ) = ( ( cos ‘ 𝐴 ) / ( sin ‘ 𝐴 ) ) )