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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 7779 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | cosadd 11193 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℂ) → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) | |
3 | 1, 2 | mpdan 413 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
4 | negid 7826 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
5 | 4 | fveq2d 5344 | . . 3 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (cos‘0)) |
6 | cos0 11186 | . . 3 ⊢ (cos‘0) = 1 | |
7 | 5, 6 | syl6eq 2143 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = 1) |
8 | sincl 11162 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
9 | 8 | sqcld 10215 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) ∈ ℂ) |
10 | coscl 11163 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
11 | 10 | sqcld 10215 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) ∈ ℂ) |
12 | 9, 11 | addcomd 7730 | . . 3 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) |
13 | 10 | sqvald 10214 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘𝐴))) |
14 | cosneg 11183 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | |
15 | 14 | oveq2d 5706 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) = ((cos‘𝐴) · (cos‘𝐴))) |
16 | 13, 15 | eqtr4d 2130 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘-𝐴))) |
17 | 8 | sqvald 10214 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · (sin‘𝐴))) |
18 | sinneg 11182 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) | |
19 | 18 | negeqd 7774 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = --(sin‘𝐴)) |
20 | 8 | negnegd 7881 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → --(sin‘𝐴) = (sin‘𝐴)) |
21 | 19, 20 | eqtrd 2127 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = (sin‘𝐴)) |
22 | 21 | oveq2d 5706 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = ((sin‘𝐴) · (sin‘𝐴))) |
23 | 17, 22 | eqtr4d 2130 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · -(sin‘-𝐴))) |
24 | 1 | sincld 11166 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) ∈ ℂ) |
25 | 8, 24 | mulneg2d 7987 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = -((sin‘𝐴) · (sin‘-𝐴))) |
26 | 23, 25 | eqtrd 2127 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = -((sin‘𝐴) · (sin‘-𝐴))) |
27 | 16, 26 | oveq12d 5708 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴)))) |
28 | 1 | coscld 11167 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) ∈ ℂ) |
29 | 10, 28 | mulcld 7605 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) ∈ ℂ) |
30 | 8, 24 | mulcld 7605 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · (sin‘-𝐴)) ∈ ℂ) |
31 | 29, 30 | negsubd 7896 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴))) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
32 | 12, 27, 31 | 3eqtrrd 2132 | . 2 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴))) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
33 | 3, 7, 32 | 3eqtr3rd 2136 | 1 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 ‘cfv 5049 (class class class)co 5690 ℂcc 7445 0cc0 7447 1c1 7448 + caddc 7450 · cmul 7452 − cmin 7750 -cneg 7751 2c2 8571 ↑cexp 10085 sincsin 11099 cosccos 11100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 ax-caucvg 7562 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-disj 3845 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-isom 5058 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-frec 6194 df-1o 6219 df-oadd 6223 df-er 6332 df-en 6538 df-dom 6539 df-fin 6540 df-sup 6759 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-n0 8772 df-z 8849 df-uz 9119 df-q 9204 df-rp 9234 df-ico 9460 df-fz 9574 df-fzo 9703 df-iseq 10002 df-seq3 10003 df-exp 10086 df-fac 10265 df-bc 10287 df-ihash 10315 df-cj 10407 df-re 10408 df-im 10409 df-rsqrt 10562 df-abs 10563 df-clim 10838 df-sumdc 10913 df-ef 11103 df-sin 11105 df-cos 11106 |
This theorem is referenced by: cos2t 11206 cos2tsin 11207 sinbnd 11208 cosbnd 11209 absefi 11223 |
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