MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cos2tsin Unicode version

Theorem cos2tsin 12743
Description: Double-angle formula for cosine in terms of sine. (Contributed by NM, 12-Sep-2008.)
Assertion
Ref Expression
cos2tsin  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( 1  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) ) )

Proof of Theorem cos2tsin
StepHypRef Expression
1 cos2t 12742 . 2  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( ( 2  x.  ( ( cos `  A
) ^ 2 ) )  -  1 ) )
2 sincl 12690 . . . . . . . 8  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
32sqcld 11484 . . . . . . 7  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
4 coscl 12691 . . . . . . . 8  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
54sqcld 11484 . . . . . . 7  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
6 2cn 10034 . . . . . . . 8  |-  2  e.  CC
7 adddi 9043 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  ( ( sin `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  e.  CC )  -> 
( 2  x.  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )  =  ( ( 2  x.  ( ( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) ) )
86, 7mp3an1 1266 . . . . . . 7  |-  ( ( ( ( sin `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  e.  CC )  -> 
( 2  x.  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )  =  ( ( 2  x.  ( ( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) ) )
93, 5, 8syl2anc 643 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  ( ( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )  =  ( ( 2  x.  ( ( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) ) )
10 sincossq 12740 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
1110oveq2d 6064 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  ( ( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )  =  ( 2  x.  1 ) )
129, 11eqtr3d 2446 . . . . 5  |-  ( A  e.  CC  ->  (
( 2  x.  (
( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) )  =  ( 2  x.  1 ) )
136mulid1i 9056 . . . . 5  |-  ( 2  x.  1 )  =  2
1412, 13syl6eq 2460 . . . 4  |-  ( A  e.  CC  ->  (
( 2  x.  (
( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) )  =  2 )
15 mulcl 9038 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( ( sin `  A
) ^ 2 )  e.  CC )  -> 
( 2  x.  (
( sin `  A
) ^ 2 ) )  e.  CC )
166, 3, 15sylancr 645 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( ( sin `  A ) ^ 2 ) )  e.  CC )
17 mulcl 9038 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( ( cos `  A
) ^ 2 )  e.  CC )  -> 
( 2  x.  (
( cos `  A
) ^ 2 ) )  e.  CC )
186, 5, 17sylancr 645 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( ( cos `  A ) ^ 2 ) )  e.  CC )
19 subadd 9272 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( 2  x.  (
( sin `  A
) ^ 2 ) )  e.  CC  /\  ( 2  x.  (
( cos `  A
) ^ 2 ) )  e.  CC )  ->  ( ( 2  -  ( 2  x.  ( ( sin `  A
) ^ 2 ) ) )  =  ( 2  x.  ( ( cos `  A ) ^ 2 ) )  <-> 
( ( 2  x.  ( ( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) )  =  2 ) )
206, 19mp3an1 1266 . . . . 5  |-  ( ( ( 2  x.  (
( sin `  A
) ^ 2 ) )  e.  CC  /\  ( 2  x.  (
( cos `  A
) ^ 2 ) )  e.  CC )  ->  ( ( 2  -  ( 2  x.  ( ( sin `  A
) ^ 2 ) ) )  =  ( 2  x.  ( ( cos `  A ) ^ 2 ) )  <-> 
( ( 2  x.  ( ( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) )  =  2 ) )
2116, 18, 20syl2anc 643 . . . 4  |-  ( A  e.  CC  ->  (
( 2  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) )  =  ( 2  x.  ( ( cos `  A ) ^ 2 ) )  <->  ( (
2  x.  ( ( sin `  A ) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A
) ^ 2 ) ) )  =  2 ) )
2214, 21mpbird 224 . . 3  |-  ( A  e.  CC  ->  (
2  -  ( 2  x.  ( ( sin `  A ) ^ 2 ) ) )  =  ( 2  x.  (
( cos `  A
) ^ 2 ) ) )
2322oveq1d 6063 . 2  |-  ( A  e.  CC  ->  (
( 2  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) )  -  1 )  =  ( ( 2  x.  ( ( cos `  A ) ^ 2 ) )  -  1 ) )
24 ax-1cn 9012 . . . . 5  |-  1  e.  CC
25 sub32 9299 . . . . 5  |-  ( ( 2  e.  CC  /\  ( 2  x.  (
( sin `  A
) ^ 2 ) )  e.  CC  /\  1  e.  CC )  ->  ( ( 2  -  ( 2  x.  (
( sin `  A
) ^ 2 ) ) )  -  1 )  =  ( ( 2  -  1 )  -  ( 2  x.  ( ( sin `  A
) ^ 2 ) ) ) )
266, 24, 25mp3an13 1270 . . . 4  |-  ( ( 2  x.  ( ( sin `  A ) ^ 2 ) )  e.  CC  ->  (
( 2  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) )  -  1 )  =  ( ( 2  -  1 )  -  ( 2  x.  (
( sin `  A
) ^ 2 ) ) ) )
2716, 26syl 16 . . 3  |-  ( A  e.  CC  ->  (
( 2  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) )  -  1 )  =  ( ( 2  -  1 )  -  ( 2  x.  (
( sin `  A
) ^ 2 ) ) ) )
28 2m1e1 10059 . . . 4  |-  ( 2  -  1 )  =  1
2928oveq1i 6058 . . 3  |-  ( ( 2  -  1 )  -  ( 2  x.  ( ( sin `  A
) ^ 2 ) ) )  =  ( 1  -  ( 2  x.  ( ( sin `  A ) ^ 2 ) ) )
3027, 29syl6eq 2460 . 2  |-  ( A  e.  CC  ->  (
( 2  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) )  -  1 )  =  ( 1  -  ( 2  x.  (
( sin `  A
) ^ 2 ) ) ) )
311, 23, 303eqtr2d 2450 1  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( 1  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048   CCcc 8952   1c1 8955    + caddc 8957    x. cmul 8959    - cmin 9255   2c2 10013   ^cexp 11345   sincsin 12629   cosccos 12630
This theorem is referenced by:  coseq1  20391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-ico 10886  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-shft 11845  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246  df-sum 12443  df-ef 12633  df-sin 12635  df-cos 12636
  Copyright terms: Public domain W3C validator