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Showing content from https://dlmf.nist.gov/4.14 below:

§4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

§4.14 Definitions and Periodicity 4.14.1 sin ⁡ z = e i ⁢ z − e − i ⁢ z 2 ⁢ i , 4.14.2 cos ⁡ z = e i ⁢ z + e − i ⁢ z 2 , 4.14.3 cos ⁡ z ± i ⁢ sin ⁡ z = e ± i ⁢ z , 4.14.4 tan ⁡ z = sin ⁡ z cos ⁡ z , 4.14.5 csc ⁡ z = 1 sin ⁡ z , 4.14.6 sec ⁡ z = 1 cos ⁡ z , 4.14.7 cot ⁡ z = cos ⁡ z sin ⁡ z = 1 tan ⁡ z .

The functions sin ⁡ z and cos ⁡ z are entire. In ℂ the zeros of sin ⁡ z are z = k ⁢ π , k ∈ ℤ ; the zeros of cos ⁡ z are z = ( k + 1 2 ) ⁢ π , k ∈ ℤ . The functions tan ⁡ z , csc ⁡ z , sec ⁡ z , and cot ⁡ z are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7).

For k ∈ ℤ

4.14.8 sin ⁡ ( z + 2 ⁢ k ⁢ π ) = sin ⁡ z , 4.14.9 cos ⁡ ( z + 2 ⁢ k ⁢ π ) = cos ⁡ z , 4.14.10 tan ⁡ ( z + k ⁢ π ) = tan ⁡ z .

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