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Theorem cos2t 12458
Description: Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.)
Assertion
Ref Expression
cos2t  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( ( 2  x.  ( ( cos `  A
) ^ 2 ) )  -  1 ) )

Proof of Theorem cos2t
StepHypRef Expression
1 coscl 12407 . . . 4  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
21sqcld 11243 . . 3  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
3 ax-1cn 8795 . . . 4  |-  1  e.  CC
4 subsub3 9079 . . . 4  |-  ( ( ( ( cos `  A
) ^ 2 )  e.  CC  /\  1  e.  CC  /\  ( ( cos `  A ) ^ 2 )  e.  CC )  ->  (
( ( cos `  A
) ^ 2 )  -  ( 1  -  ( ( cos `  A
) ^ 2 ) ) )  =  ( ( ( ( cos `  A ) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  -  1 ) )
53, 4mp3an2 1265 . . 3  |-  ( ( ( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  e.  CC )  -> 
( ( ( cos `  A ) ^ 2 )  -  ( 1  -  ( ( cos `  A ) ^ 2 ) ) )  =  ( ( ( ( cos `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  -  1 ) )
62, 2, 5syl2anc 642 . 2  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  -  ( 1  -  ( ( cos `  A
) ^ 2 ) ) )  =  ( ( ( ( cos `  A ) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  -  1 ) )
7 cosadd 12445 . . . . 5  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( cos `  ( A  +  A )
)  =  ( ( ( cos `  A
)  x.  ( cos `  A ) )  -  ( ( sin `  A
)  x.  ( sin `  A ) ) ) )
87anidms 626 . . . 4  |-  ( A  e.  CC  ->  ( cos `  ( A  +  A ) )  =  ( ( ( cos `  A )  x.  ( cos `  A ) )  -  ( ( sin `  A )  x.  ( sin `  A ) ) ) )
9 2times 9843 . . . . 5  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
109fveq2d 5529 . . . 4  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( cos `  ( A  +  A )
) )
111sqvald 11242 . . . . 5  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  =  ( ( cos `  A )  x.  ( cos `  A ) ) )
12 sincl 12406 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
1312sqvald 11242 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  ( ( sin `  A )  x.  ( sin `  A ) ) )
1411, 13oveq12d 5876 . . . 4  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  -  ( ( sin `  A ) ^ 2 ) )  =  ( ( ( cos `  A
)  x.  ( cos `  A ) )  -  ( ( sin `  A
)  x.  ( sin `  A ) ) ) )
158, 10, 143eqtr4d 2325 . . 3  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( ( ( cos `  A ) ^ 2 )  -  ( ( sin `  A ) ^ 2 ) ) )
1612sqcld 11243 . . . . . . 7  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
1716, 2addcomd 9014 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) ) )
18 sincossq 12456 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
1917, 18eqtr3d 2317 . . . . 5  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  1 )
20 subadd 9054 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( sin `  A
) ^ 2 )  e.  CC )  -> 
( ( 1  -  ( ( cos `  A
) ^ 2 ) )  =  ( ( sin `  A ) ^ 2 )  <->  ( (
( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  1 ) )
213, 20mp3an1 1264 . . . . . 6  |-  ( ( ( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( sin `  A
) ^ 2 )  e.  CC )  -> 
( ( 1  -  ( ( cos `  A
) ^ 2 ) )  =  ( ( sin `  A ) ^ 2 )  <->  ( (
( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  1 ) )
222, 16, 21syl2anc 642 . . . . 5  |-  ( A  e.  CC  ->  (
( 1  -  (
( cos `  A
) ^ 2 ) )  =  ( ( sin `  A ) ^ 2 )  <->  ( (
( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  1 ) )
2319, 22mpbird 223 . . . 4  |-  ( A  e.  CC  ->  (
1  -  ( ( cos `  A ) ^ 2 ) )  =  ( ( sin `  A ) ^ 2 ) )
2423oveq2d 5874 . . 3  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  -  ( 1  -  ( ( cos `  A
) ^ 2 ) ) )  =  ( ( ( cos `  A
) ^ 2 )  -  ( ( sin `  A ) ^ 2 ) ) )
2515, 24eqtr4d 2318 . 2  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( ( ( cos `  A ) ^ 2 )  -  ( 1  -  ( ( cos `  A ) ^ 2 ) ) ) )
2622timesd 9954 . . 3  |-  ( A  e.  CC  ->  (
2  x.  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( cos `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) ) )
2726oveq1d 5873 . 2  |-  ( A  e.  CC  ->  (
( 2  x.  (
( cos `  A
) ^ 2 ) )  -  1 )  =  ( ( ( ( cos `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  -  1 ) )
286, 25, 273eqtr4d 2325 1  |-  ( A  e.  CC  ->  ( cos `  ( 2  x.  A ) )  =  ( ( 2  x.  ( ( cos `  A
) ^ 2 ) )  -  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   2c2 9795   ^cexp 11104   sincsin 12345   cosccos 12346
This theorem is referenced by:  cos2tsin  12459  cos2bnd  12468  cospi  19840  cos2pi  19844  tangtx  19873  coskpi  19888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352
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