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Theorem sinmul 11747
Description: Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11740 and cossub 11744. (Contributed by David A. Wheeler, 26-May-2015.)
Assertion
Ref Expression
sinmul  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A
)  x.  ( sin `  B ) )  =  ( ( ( cos `  ( A  -  B
) )  -  ( cos `  ( A  +  B ) ) )  /  2 ) )

Proof of Theorem sinmul
StepHypRef Expression
1 cossub 11744 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  -  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
2 cosadd 11740 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
31, 2oveq12d 5892 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  ( A  -  B )
)  -  ( cos `  ( A  +  B
) ) )  =  ( ( ( ( cos `  A )  x.  ( cos `  B
) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) )  -  ( ( ( cos `  A )  x.  ( cos `  B
) )  -  (
( sin `  A
)  x.  ( sin `  B ) ) ) ) )
4 coscl 11710 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
5 coscl 11710 . . . . . 6  |-  ( B  e.  CC  ->  ( cos `  B )  e.  CC )
6 mulcl 7937 . . . . . 6  |-  ( ( ( cos `  A
)  e.  CC  /\  ( cos `  B )  e.  CC )  -> 
( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC )
74, 5, 6syl2an 289 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC )
8 sincl 11709 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
9 sincl 11709 . . . . . 6  |-  ( B  e.  CC  ->  ( sin `  B )  e.  CC )
10 mulcl 7937 . . . . . 6  |-  ( ( ( sin `  A
)  e.  CC  /\  ( sin `  B )  e.  CC )  -> 
( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC )
118, 9, 10syl2an 289 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC )
12 pnncan 8196 . . . . . . 7  |-  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC  /\  ( ( sin `  A )  x.  ( sin `  B
) )  e.  CC  /\  ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC )  ->  (
( ( ( cos `  A )  x.  ( cos `  B ) )  +  ( ( sin `  A )  x.  ( sin `  B ) ) )  -  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )  =  ( ( ( sin `  A
)  x.  ( sin `  B ) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
13123anidm23 1297 . . . . . 6  |-  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC  /\  ( ( sin `  A )  x.  ( sin `  B
) )  e.  CC )  ->  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) )  -  ( ( ( cos `  A )  x.  ( cos `  B
) )  -  (
( sin `  A
)  x.  ( sin `  B ) ) ) )  =  ( ( ( sin `  A
)  x.  ( sin `  B ) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
14 2times 9045 . . . . . . 7  |-  ( ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC  ->  ( 2  x.  ( ( sin `  A )  x.  ( sin `  B ) ) )  =  ( ( ( sin `  A
)  x.  ( sin `  B ) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
1514adantl 277 . . . . . 6  |-  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC  /\  ( ( sin `  A )  x.  ( sin `  B
) )  e.  CC )  ->  ( 2  x.  ( ( sin `  A
)  x.  ( sin `  B ) ) )  =  ( ( ( sin `  A )  x.  ( sin `  B
) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
1613, 15eqtr4d 2213 . . . . 5  |-  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC  /\  ( ( sin `  A )  x.  ( sin `  B
) )  e.  CC )  ->  ( ( ( ( cos `  A
)  x.  ( cos `  B ) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) )  -  ( ( ( cos `  A )  x.  ( cos `  B
) )  -  (
( sin `  A
)  x.  ( sin `  B ) ) ) )  =  ( 2  x.  ( ( sin `  A )  x.  ( sin `  B ) ) ) )
177, 11, 16syl2anc 411 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( cos `  A )  x.  ( cos `  B
) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) )  -  ( ( ( cos `  A )  x.  ( cos `  B
) )  -  (
( sin `  A
)  x.  ( sin `  B ) ) ) )  =  ( 2  x.  ( ( sin `  A )  x.  ( sin `  B ) ) ) )
18 2cn 8988 . . . . 5  |-  2  e.  CC
19 mulcom 7939 . . . . 5  |-  ( ( 2  e.  CC  /\  ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC )  ->  (
2  x.  ( ( sin `  A )  x.  ( sin `  B
) ) )  =  ( ( ( sin `  A )  x.  ( sin `  B ) )  x.  2 ) )
2018, 11, 19sylancr 414 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  (
( sin `  A
)  x.  ( sin `  B ) ) )  =  ( ( ( sin `  A )  x.  ( sin `  B
) )  x.  2 ) )
213, 17, 203eqtrd 2214 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  ( A  -  B )
)  -  ( cos `  ( A  +  B
) ) )  =  ( ( ( sin `  A )  x.  ( sin `  B ) )  x.  2 ) )
2221oveq1d 5889 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( cos `  ( A  -  B
) )  -  ( cos `  ( A  +  B ) ) )  /  2 )  =  ( ( ( ( sin `  A )  x.  ( sin `  B
) )  x.  2 )  /  2 ) )
23 2ap0 9010 . . . 4  |-  2 #  0
24 divcanap4 8654 . . . 4  |-  ( ( ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC  /\  2  e.  CC  /\  2 #  0 )  ->  ( (
( ( sin `  A
)  x.  ( sin `  B ) )  x.  2 )  /  2
)  =  ( ( sin `  A )  x.  ( sin `  B
) ) )
2518, 23, 24mp3an23 1329 . . 3  |-  ( ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC  ->  ( (
( ( sin `  A
)  x.  ( sin `  B ) )  x.  2 )  /  2
)  =  ( ( sin `  A )  x.  ( sin `  B
) ) )
2611, 25syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( sin `  A )  x.  ( sin `  B
) )  x.  2 )  /  2 )  =  ( ( sin `  A )  x.  ( sin `  B ) ) )
2722, 26eqtr2d 2211 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A
)  x.  ( sin `  B ) )  =  ( ( ( cos `  ( A  -  B
) )  -  ( cos `  ( A  +  B ) ) )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   0cc0 7810    + caddc 7813    x. cmul 7815    - cmin 8126   # cap 8536    / cdiv 8627   2c2 8968   sincsin 11647   cosccos 11648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-disj 3981  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-sup 6982  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-ico 9892  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-fac 10701  df-bc 10723  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-sumdc 11357  df-ef 11651  df-sin 11653  df-cos 11654
This theorem is referenced by:  ptolemy  14138
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