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 e. CC /\ K e. ZZ ) -> ( cos ` ( A + ( K x. ( 2 x. _pi ) ) ) ) = ( cos ` A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cosval | |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) ) |
|
| 2 | cosval | |- ( ( A + ( K x. ( 2 x. _pi ) ) ) e. CC -> ( cos ` ( A + ( K x. ( 2 x. _pi ) ) ) ) = ( ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. _pi ) ) ) ) ) + ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. _pi ) ) ) ) ) ) / 2 ) ) |
|
| 3 | 1 2 | sinperlem | |- ( ( A e. CC /\ K e. ZZ ) -> ( cos ` ( A + ( K x. ( 2 x. _pi ) ) ) ) = ( cos ` A ) ) |