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Theorem sincossq 12777
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 9306 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 cosadd 12766 . . 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 650 . 2  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  ( ( ( cos `  A )  x.  ( cos `  -u A ) )  -  ( ( sin `  A )  x.  ( sin `  -u A ) ) ) )
4 negid 9348 . . . 4  |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
54fveq2d 5732 . . 3  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  ( cos `  0
) )
6 cos0 12751 . . 3  |-  ( cos `  0 )  =  1
75, 6syl6eq 2484 . 2  |-  ( A  e.  CC  ->  ( cos `  ( A  +  -u A ) )  =  1 )
8 sincl 12727 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
98sqcld 11521 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
10 coscl 12728 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
1110sqcld 11521 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
129, 11addcomd 9268 . . 3  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) ) )
1310sqvald 11520 . . . . 5  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  =  ( ( cos `  A )  x.  ( cos `  A ) ) )
14 cosneg 12748 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  -u A )  =  ( cos `  A
) )
1514oveq2d 6097 . . . . 5  |-  ( A  e.  CC  ->  (
( cos `  A
)  x.  ( cos `  -u A ) )  =  ( ( cos `  A )  x.  ( cos `  A ) ) )
1613, 15eqtr4d 2471 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  =  ( ( cos `  A )  x.  ( cos `  -u A ) ) )
178sqvald 11520 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  ( ( sin `  A )  x.  ( sin `  A ) ) )
18 sinneg 12747 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( sin `  -u A )  = 
-u ( sin `  A
) )
1918negeqd 9300 . . . . . . . 8  |-  ( A  e.  CC  ->  -u ( sin `  -u A )  = 
-u -u ( sin `  A
) )
208negnegd 9402 . . . . . . . 8  |-  ( A  e.  CC  ->  -u -u ( sin `  A )  =  ( sin `  A
) )
2119, 20eqtrd 2468 . . . . . . 7  |-  ( A  e.  CC  ->  -u ( sin `  -u A )  =  ( sin `  A
) )
2221oveq2d 6097 . . . . . 6  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  -u ( sin `  -u A ) )  =  ( ( sin `  A )  x.  ( sin `  A ) ) )
2317, 22eqtr4d 2471 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  ( ( sin `  A )  x.  -u ( sin `  -u A ) ) )
241sincld 12731 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  -u A )  e.  CC )
258, 24mulneg2d 9487 . . . . 5  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  -u ( sin `  -u A ) )  =  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) )
2623, 25eqtrd 2468 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  =  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) )
2716, 26oveq12d 6099 . . 3  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  ( ( ( cos `  A
)  x.  ( cos `  -u A ) )  +  -u ( ( sin `  A )  x.  ( sin `  -u A ) ) ) )
281coscld 12732 . . . . 5  |-  ( A  e.  CC  ->  ( cos `  -u A )  e.  CC )
2910, 28mulcld 9108 . . . 4  |-  ( A  e.  CC  ->  (
( cos `  A
)  x.  ( cos `  -u A ) )  e.  CC )
308, 24mulcld 9108 . . . 4  |-  ( A  e.  CC  ->  (
( sin `  A
)  x.  ( sin `  -u A ) )  e.  CC )
3129, 30negsubd 9417 . . 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 2473 . 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 2477 1  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292   2c2 10049   ^cexp 11382   sincsin 12666   cosccos 12667
This theorem is referenced by:  cos2t  12779  cos2tsin  12780  sinbnd  12781  cosbnd  12782  absefi  12797  sinhalfpilem  20374  sincos6thpi  20423  efif1olem4  20447  asinsin  20732  atandmtan  20760  basellem8  20870  itgsinexp  27725  onetansqsecsq  28504  cotsqcscsq  28505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673
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