Theorem cosper

Description: The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008) (Revised by Mario Carneiro, 10-May-2014)

	Ref	Expression
Assertion	cosper	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ ) → ( cos ‘ ( 𝐴 + ( 𝐾 · ( 2 · π ) ) ) ) = ( cos ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		cosval	⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) = ( ( ( exp ‘ ( i · 𝐴 ) ) + ( exp ‘ ( - i · 𝐴 ) ) ) / 2 ) )
2		cosval	⊢ ( ( 𝐴 + ( 𝐾 · ( 2 · π ) ) ) ∈ ℂ → ( cos ‘ ( 𝐴 + ( 𝐾 · ( 2 · π ) ) ) ) = ( ( ( exp ‘ ( i · ( 𝐴 + ( 𝐾 · ( 2 · π ) ) ) ) ) + ( exp ‘ ( - i · ( 𝐴 + ( 𝐾 · ( 2 · π ) ) ) ) ) ) / 2 ) )
3	1 2	sinperlem	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ ) → ( cos ‘ ( 𝐴 + ( 𝐾 · ( 2 · π ) ) ) ) = ( cos ‘ 𝐴 ) )