Theorem recsec

Description: The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014)

	Ref	Expression
Assertion	recsec	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) = ( 1 / ( sec ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		secval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( sec ‘ 𝐴 ) = ( 1 / ( cos ‘ 𝐴 ) ) )
2	1	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( 1 / ( sec ‘ 𝐴 ) ) = ( 1 / ( 1 / ( cos ‘ 𝐴 ) ) ) )
3		coscl	⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ )
4		recrec	⊢ ( ( ( cos ‘ 𝐴 ) ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( 1 / ( 1 / ( cos ‘ 𝐴 ) ) ) = ( cos ‘ 𝐴 ) )
5	3 4	sylan	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( 1 / ( 1 / ( cos ‘ 𝐴 ) ) ) = ( cos ‘ 𝐴 ) )
6	2 5	eqtr2d	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) = ( 1 / ( sec ‘ 𝐴 ) ) )