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Theorem cos2tsin 12785
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 12784 . 2  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( ( 2  x.  ( ( cos `  A
) ^ 2 ) )  -  1 ) )
2 sincl 12732 . . . . . . . 8  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
32sqcld 11526 . . . . . . 7  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
4 coscl 12733 . . . . . . . 8  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
54sqcld 11526 . . . . . . 7  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
6 2cn 10075 . . . . . . . 8  |-  2  e.  CC
7 adddi 9084 . . . . . . . 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 1267 . . . . . . 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 644 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  ( ( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )  =  ( ( 2  x.  ( ( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) ) )
10 sincossq 12782 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
1110oveq2d 6100 . . . . . 6  |-  ( A  e.  CC  ->  (
2  x.  ( ( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )  =  ( 2  x.  1 ) )
129, 11eqtr3d 2472 . . . . 5  |-  ( A  e.  CC  ->  (
( 2  x.  (
( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) )  =  ( 2  x.  1 ) )
136mulid1i 9097 . . . . 5  |-  ( 2  x.  1 )  =  2
1412, 13syl6eq 2486 . . . 4  |-  ( A  e.  CC  ->  (
( 2  x.  (
( sin `  A
) ^ 2 ) )  +  ( 2  x.  ( ( cos `  A ) ^ 2 ) ) )  =  2 )
15 mulcl 9079 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( ( sin `  A
) ^ 2 )  e.  CC )  -> 
( 2  x.  (
( sin `  A
) ^ 2 ) )  e.  CC )
166, 3, 15sylancr 646 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( ( sin `  A ) ^ 2 ) )  e.  CC )
17 mulcl 9079 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( ( cos `  A
) ^ 2 )  e.  CC )  -> 
( 2  x.  (
( cos `  A
) ^ 2 ) )  e.  CC )
186, 5, 17sylancr 646 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  ( ( cos `  A ) ^ 2 ) )  e.  CC )
19 subadd 9313 . . . . . 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 1267 . . . . 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 644 . . . 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 225 . . 3  |-  ( A  e.  CC  ->  (
2  -  ( 2  x.  ( ( sin `  A ) ^ 2 ) ) )  =  ( 2  x.  (
( cos `  A
) ^ 2 ) ) )
2322oveq1d 6099 . 2  |-  ( A  e.  CC  ->  (
( 2  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) )  -  1 )  =  ( ( 2  x.  ( ( cos `  A ) ^ 2 ) )  -  1 ) )
24 ax-1cn 9053 . . . . 5  |-  1  e.  CC
25 sub32 9340 . . . . 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 1271 . . . 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 10100 . . . 4  |-  ( 2  -  1 )  =  1
2928oveq1i 6094 . . 3  |-  ( ( 2  -  1 )  -  ( 2  x.  ( ( sin `  A
) ^ 2 ) ) )  =  ( 1  -  ( 2  x.  ( ( sin `  A ) ^ 2 ) ) )
3027, 29syl6eq 2486 . 2  |-  ( A  e.  CC  ->  (
( 2  -  (
2  x.  ( ( sin `  A ) ^ 2 ) ) )  -  1 )  =  ( 1  -  ( 2  x.  (
( sin `  A
) ^ 2 ) ) ) )
311, 23, 303eqtr2d 2476 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 178    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   CCcc 8993   1c1 8996    + caddc 8998    x. cmul 9000    - cmin 9296   2c2 10054   ^cexp 11387   sincsin 12671   cosccos 12672
This theorem is referenced by:  coseq1  20435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-ico 10927  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678
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