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Theorem sincossq 13582
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 9725 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 cosadd 13571 . . 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 9771 . . . 4  |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
54fveq2d 5806 . . 3  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  ( cos `  0
) )
6 cos0 13556 . . 3  |-  ( cos `  0 )  =  1
75, 6syl6eq 2511 . 2  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  1 )
8 sincl 13532 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
98sqcld 12127 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
10 coscl 13533 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
1110sqcld 12127 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
129, 11addcomd 9686 . . 3  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) ) )
1310sqvald 12126 . . . . 5  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  =  ( ( cos `  A )  x.  ( cos `  A ) ) )
14 cosneg 13553 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  -u A )  =  ( cos `  A
) )
1514oveq2d 6219 . . . . 5  |-  ( A  e.  CC  ->  (
( cos `  A
)  x.  ( cos `  -u A ) )  =  ( ( cos `  A )  x.  ( cos `  A ) ) )
1613, 15eqtr4d 2498 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  =  ( ( cos `  A )  x.  ( cos `  -u A ) ) )
178sqvald 12126 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  ( ( sin `  A )  x.  ( sin `  A ) ) )
18 sinneg 13552 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( sin `  -u A )  = 
-u ( sin `  A
) )
1918negeqd 9719 . . . . . . . 8  |-  ( A  e.  CC  ->  -u ( sin `  -u A )  = 
-u -u ( sin `  A
) )
208negnegd 9825 . . . . . . . 8  |-  ( A  e.  CC  ->  -u -u ( sin `  A )  =  ( sin `  A
) )
2119, 20eqtrd 2495 . . . . . . 7  |-  ( A  e.  CC  ->  -u ( sin `  -u A )  =  ( sin `  A
) )
2221oveq2d 6219 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  -u ( sin `  -u A ) )  =  ( ( sin `  A )  x.  ( sin `  A ) ) )
2317, 22eqtr4d 2498 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  ( ( sin `  A )  x.  -u ( sin `  -u A ) ) )
241sincld 13536 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  -u A )  e.  CC )
258, 24mulneg2d 9913 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  -u ( sin `  -u A ) )  =  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) )
2623, 25eqtrd 2495 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) )
2716, 26oveq12d 6221 . . 3  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  ( ( ( cos `  A
)  x.  ( cos `  -u A ) )  +  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) ) )
281coscld 13537 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  -u A )  e.  CC )
2910, 28mulcld 9521 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
)  x.  ( cos `  -u A ) )  e.  CC )
308, 24mulcld 9521 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  ( sin `  -u A ) )  e.  CC )
3129, 30negsubd 9840 . . 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 2500 . 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 2504 1  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   CCcc 9395   0cc0 9397   1c1 9398    + caddc 9400    x. cmul 9402    - cmin 9710   -ucneg 9711   2c2 10486   ^cexp 11986   sincsin 13471   cosccos 13472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-addf 9476  ax-mulf 9477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-ico 11421  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-fac 12173  df-bc 12200  df-hash 12225  df-shft 12678  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-limsup 13071  df-clim 13088  df-rlim 13089  df-sum 13286  df-ef 13475  df-sin 13477  df-cos 13478
This theorem is referenced by:  cos2t  13584  cos2tsin  13585  sinbnd  13586  cosbnd  13587  absefi  13602  sinhalfpilem  22068  sincos6thpi  22120  efif1olem4  22144  heron  22376  asinsin  22430  atandmtan  22458  basellem8  22568  sin2h  28593  tan2h  28595  dvtan  28613  itgsinexp  29966  onetansqsecsq  31447  cotsqcscsq  31448
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