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Theorem sincossq 13788
Description: Sine squared plus cosine squared is 1. Equation 17 of [Gleason] p. 311. Note that this holds for non-real arguments, even though individually each term is unbounded. (Contributed by NM, 15-Jan-2006.)
Assertion
Ref Expression
sincossq  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )

Proof of Theorem sincossq
StepHypRef Expression
1 negcl 9825 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 cosadd 13777 . . 3  |-  ( ( A  e.  CC  /\  -u A  e.  CC )  ->  ( cos `  ( A  +  -u A ) )  =  ( ( ( cos `  A
)  x.  ( cos `  -u A ) )  -  ( ( sin `  A )  x.  ( sin `  -u A ) ) ) )
31, 2mpdan 668 . 2  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  ( ( ( cos `  A )  x.  ( cos `  -u A ) )  -  ( ( sin `  A )  x.  ( sin `  -u A ) ) ) )
4 negid 9871 . . . 4  |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
54fveq2d 5860 . . 3  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  ( cos `  0
) )
6 cos0 13762 . . 3  |-  ( cos `  0 )  =  1
75, 6syl6eq 2500 . 2  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  1 )
8 sincl 13738 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
98sqcld 12287 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
10 coscl 13739 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
1110sqcld 12287 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
129, 11addcomd 9785 . . 3  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) ) )
1310sqvald 12286 . . . . 5  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  =  ( ( cos `  A )  x.  ( cos `  A ) ) )
14 cosneg 13759 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  -u A )  =  ( cos `  A
) )
1514oveq2d 6297 . . . . 5  |-  ( A  e.  CC  ->  (
( cos `  A
)  x.  ( cos `  -u A ) )  =  ( ( cos `  A )  x.  ( cos `  A ) ) )
1613, 15eqtr4d 2487 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  =  ( ( cos `  A )  x.  ( cos `  -u A ) ) )
178sqvald 12286 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  ( ( sin `  A )  x.  ( sin `  A ) ) )
18 sinneg 13758 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( sin `  -u A )  = 
-u ( sin `  A
) )
1918negeqd 9819 . . . . . . . 8  |-  ( A  e.  CC  ->  -u ( sin `  -u A )  = 
-u -u ( sin `  A
) )
208negnegd 9927 . . . . . . . 8  |-  ( A  e.  CC  ->  -u -u ( sin `  A )  =  ( sin `  A
) )
2119, 20eqtrd 2484 . . . . . . 7  |-  ( A  e.  CC  ->  -u ( sin `  -u A )  =  ( sin `  A
) )
2221oveq2d 6297 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  -u ( sin `  -u A ) )  =  ( ( sin `  A )  x.  ( sin `  A ) ) )
2317, 22eqtr4d 2487 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  ( ( sin `  A )  x.  -u ( sin `  -u A ) ) )
241sincld 13742 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  -u A )  e.  CC )
258, 24mulneg2d 10016 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  -u ( sin `  -u A ) )  =  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) )
2623, 25eqtrd 2484 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) )
2716, 26oveq12d 6299 . . 3  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  ( ( ( cos `  A
)  x.  ( cos `  -u A ) )  +  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) ) )
281coscld 13743 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  -u A )  e.  CC )
2910, 28mulcld 9619 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
)  x.  ( cos `  -u A ) )  e.  CC )
308, 24mulcld 9619 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  ( sin `  -u A ) )  e.  CC )
3129, 30negsubd 9942 . . 3  |-  ( A  e.  CC  ->  (
( ( cos `  A
)  x.  ( cos `  -u A ) )  +  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) )  =  ( ( ( cos `  A
)  x.  ( cos `  -u A ) )  -  ( ( sin `  A )  x.  ( sin `  -u A ) ) ) )
3212, 27, 313eqtrrd 2489 . 2  |-  ( A  e.  CC  ->  (
( ( cos `  A
)  x.  ( cos `  -u A ) )  -  ( ( sin `  A )  x.  ( sin `  -u A ) ) )  =  ( ( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) ) )
333, 7, 323eqtr3rd 2493 1  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811   2c2 10591   ^cexp 12145   sincsin 13677   cosccos 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-ico 11544  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684
This theorem is referenced by:  cos2t  13790  cos2tsin  13791  sinbnd  13792  cosbnd  13793  absefi  13808  sinhalfpilem  22728  sincos6thpi  22780  efif1olem4  22804  heron  23041  asinsin  23095  atandmtan  23123  basellem8  23233  sin2h  30020  tan2h  30022  dvtan  30040  itgsinexp  31643  onetansqsecsq  32890  cotsqcscsq  32891
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