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Mirrors > Home > ILE Home > Th. List > sincossq | GIF version |
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.) |
Ref | Expression |
---|---|
sincossq | ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 7962 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | cosadd 11444 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℂ) → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) | |
3 | 1, 2 | mpdan 417 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
4 | negid 8009 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
5 | 4 | fveq2d 5425 | . . 3 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (cos‘0)) |
6 | cos0 11437 | . . 3 ⊢ (cos‘0) = 1 | |
7 | 5, 6 | syl6eq 2188 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = 1) |
8 | sincl 11413 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
9 | 8 | sqcld 10422 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) ∈ ℂ) |
10 | coscl 11414 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
11 | 10 | sqcld 10422 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) ∈ ℂ) |
12 | 9, 11 | addcomd 7913 | . . 3 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) |
13 | 10 | sqvald 10421 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘𝐴))) |
14 | cosneg 11434 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | |
15 | 14 | oveq2d 5790 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) = ((cos‘𝐴) · (cos‘𝐴))) |
16 | 13, 15 | eqtr4d 2175 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘-𝐴))) |
17 | 8 | sqvald 10421 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · (sin‘𝐴))) |
18 | sinneg 11433 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) | |
19 | 18 | negeqd 7957 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = --(sin‘𝐴)) |
20 | 8 | negnegd 8064 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → --(sin‘𝐴) = (sin‘𝐴)) |
21 | 19, 20 | eqtrd 2172 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = (sin‘𝐴)) |
22 | 21 | oveq2d 5790 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = ((sin‘𝐴) · (sin‘𝐴))) |
23 | 17, 22 | eqtr4d 2175 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · -(sin‘-𝐴))) |
24 | 1 | sincld 11417 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) ∈ ℂ) |
25 | 8, 24 | mulneg2d 8174 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = -((sin‘𝐴) · (sin‘-𝐴))) |
26 | 23, 25 | eqtrd 2172 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = -((sin‘𝐴) · (sin‘-𝐴))) |
27 | 16, 26 | oveq12d 5792 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴)))) |
28 | 1 | coscld 11418 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) ∈ ℂ) |
29 | 10, 28 | mulcld 7786 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) ∈ ℂ) |
30 | 8, 24 | mulcld 7786 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · (sin‘-𝐴)) ∈ ℂ) |
31 | 29, 30 | negsubd 8079 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴))) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
32 | 12, 27, 31 | 3eqtrrd 2177 | . 2 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴))) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
33 | 3, 7, 32 | 3eqtr3rd 2181 | 1 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 − cmin 7933 -cneg 7934 2c2 8771 ↑cexp 10292 sincsin 11350 cosccos 11351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-bc 10494 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 df-sin 11356 df-cos 11357 |
This theorem is referenced by: cos2t 11457 cos2tsin 11458 sinbnd 11459 cosbnd 11460 absefi 11475 sinhalfpilem 12872 sincos6thpi 12923 |
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