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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 8090 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | cosadd 11668 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℂ) → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) | |
3 | 1, 2 | mpdan 418 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
4 | negid 8137 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
5 | 4 | fveq2d 5485 | . . 3 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (cos‘0)) |
6 | cos0 11661 | . . 3 ⊢ (cos‘0) = 1 | |
7 | 5, 6 | eqtrdi 2213 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = 1) |
8 | sincl 11637 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
9 | 8 | sqcld 10576 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) ∈ ℂ) |
10 | coscl 11638 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
11 | 10 | sqcld 10576 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) ∈ ℂ) |
12 | 9, 11 | addcomd 8041 | . . 3 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) |
13 | 10 | sqvald 10575 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘𝐴))) |
14 | cosneg 11658 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | |
15 | 14 | oveq2d 5853 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) = ((cos‘𝐴) · (cos‘𝐴))) |
16 | 13, 15 | eqtr4d 2200 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘-𝐴))) |
17 | 8 | sqvald 10575 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · (sin‘𝐴))) |
18 | sinneg 11657 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) | |
19 | 18 | negeqd 8085 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = --(sin‘𝐴)) |
20 | 8 | negnegd 8192 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → --(sin‘𝐴) = (sin‘𝐴)) |
21 | 19, 20 | eqtrd 2197 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = (sin‘𝐴)) |
22 | 21 | oveq2d 5853 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = ((sin‘𝐴) · (sin‘𝐴))) |
23 | 17, 22 | eqtr4d 2200 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · -(sin‘-𝐴))) |
24 | 1 | sincld 11641 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) ∈ ℂ) |
25 | 8, 24 | mulneg2d 8302 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = -((sin‘𝐴) · (sin‘-𝐴))) |
26 | 23, 25 | eqtrd 2197 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = -((sin‘𝐴) · (sin‘-𝐴))) |
27 | 16, 26 | oveq12d 5855 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴)))) |
28 | 1 | coscld 11642 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) ∈ ℂ) |
29 | 10, 28 | mulcld 7911 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) ∈ ℂ) |
30 | 8, 24 | mulcld 7911 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · (sin‘-𝐴)) ∈ ℂ) |
31 | 29, 30 | negsubd 8207 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴))) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
32 | 12, 27, 31 | 3eqtrrd 2202 | . 2 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴))) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
33 | 3, 7, 32 | 3eqtr3rd 2206 | 1 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ‘cfv 5183 (class class class)co 5837 ℂcc 7743 0cc0 7745 1c1 7746 + caddc 7748 · cmul 7750 − cmin 8061 -cneg 8062 2c2 8900 ↑cexp 10445 sincsin 11575 cosccos 11576 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 ax-arch 7864 ax-caucvg 7865 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-disj 3955 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-isom 5192 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-frec 6351 df-1o 6376 df-oadd 6380 df-er 6493 df-en 6699 df-dom 6700 df-fin 6701 df-sup 6941 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 df-inn 8850 df-2 8908 df-3 8909 df-4 8910 df-n0 9107 df-z 9184 df-uz 9459 df-q 9550 df-rp 9582 df-ico 9822 df-fz 9937 df-fzo 10069 df-seqfrec 10372 df-exp 10446 df-fac 10629 df-bc 10651 df-ihash 10679 df-cj 10774 df-re 10775 df-im 10776 df-rsqrt 10930 df-abs 10931 df-clim 11210 df-sumdc 11285 df-ef 11579 df-sin 11581 df-cos 11582 |
This theorem is referenced by: cos2t 11681 cos2tsin 11682 sinbnd 11683 cosbnd 11684 absefi 11699 sinhalfpilem 13279 sincos6thpi 13330 |
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