Description: The exponential function in terms of sine and cosine. (Contributed by NM, 14-Jan-2006)
Ref | Expression | ||
---|---|---|---|
Assertion | efmival | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn | |
|
2 | mulneg12 | |
|
3 | 1 2 | mpan | |
4 | 3 | fveq2d | |
5 | negcl | |
|
6 | efival | |
|
7 | 5 6 | syl | |
8 | cosneg | |
|
9 | sinneg | |
|
10 | 9 | oveq2d | |
11 | sincl | |
|
12 | mulneg2 | |
|
13 | 1 11 12 | sylancr | |
14 | 10 13 | eqtrd | |
15 | 8 14 | oveq12d | |
16 | coscl | |
|
17 | mulcl | |
|
18 | 1 11 17 | sylancr | |
19 | 16 18 | negsubd | |
20 | 15 19 | eqtrd | |
21 | 7 20 | eqtrd | |
22 | 4 21 | eqtrd | |