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Mirrors > Home > MPE Home > Th. List > sincossq | Structured version Visualization version 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 10739 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
2 | cosadd 15355 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐴 ∈ ℂ) → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) | |
3 | 1, 2 | mpdan 683 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
4 | negid 10787 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
5 | 4 | fveq2d 6549 | . . 3 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = (cos‘0)) |
6 | cos0 15340 | . . 3 ⊢ (cos‘0) = 1 | |
7 | 5, 6 | syl6eq 2849 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + -𝐴)) = 1) |
8 | sincl 15316 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
9 | 8 | sqcld 13362 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) ∈ ℂ) |
10 | coscl 15317 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
11 | 10 | sqcld 13362 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) ∈ ℂ) |
12 | 9, 11 | addcomd 10695 | . . 3 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) |
13 | 10 | sqvald 13361 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘𝐴))) |
14 | cosneg 15337 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) = (cos‘𝐴)) | |
15 | 14 | oveq2d 7039 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) = ((cos‘𝐴) · (cos‘𝐴))) |
16 | 13, 15 | eqtr4d 2836 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘-𝐴))) |
17 | 8 | sqvald 13361 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · (sin‘𝐴))) |
18 | sinneg 15336 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) = -(sin‘𝐴)) | |
19 | 18 | negeqd 10733 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = --(sin‘𝐴)) |
20 | 8 | negnegd 10842 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → --(sin‘𝐴) = (sin‘𝐴)) |
21 | 19, 20 | eqtrd 2833 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → -(sin‘-𝐴) = (sin‘𝐴)) |
22 | 21 | oveq2d 7039 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = ((sin‘𝐴) · (sin‘𝐴))) |
23 | 17, 22 | eqtr4d 2836 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · -(sin‘-𝐴))) |
24 | 1 | sincld 15320 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘-𝐴) ∈ ℂ) |
25 | 8, 24 | mulneg2d 10948 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · -(sin‘-𝐴)) = -((sin‘𝐴) · (sin‘-𝐴))) |
26 | 23, 25 | eqtrd 2833 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = -((sin‘𝐴) · (sin‘-𝐴))) |
27 | 16, 26 | oveq12d 7041 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴)))) |
28 | 1 | coscld 15321 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘-𝐴) ∈ ℂ) |
29 | 10, 28 | mulcld 10514 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴) · (cos‘-𝐴)) ∈ ℂ) |
30 | 8, 24 | mulcld 10514 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴) · (sin‘-𝐴)) ∈ ℂ) |
31 | 29, 30 | negsubd 10857 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) + -((sin‘𝐴) · (sin‘-𝐴))) = (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴)))) |
32 | 12, 27, 31 | 3eqtrrd 2838 | . 2 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴) · (cos‘-𝐴)) − ((sin‘𝐴) · (sin‘-𝐴))) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
33 | 3, 7, 32 | 3eqtr3rd 2842 | 1 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 0cc0 10390 1c1 10391 + caddc 10393 · cmul 10395 − cmin 10723 -cneg 10724 2c2 11546 ↑cexp 13283 sincsin 15254 cosccos 15255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-ico 12598 df-fz 12747 df-fzo 12888 df-fl 13016 df-seq 13224 df-exp 13284 df-fac 13488 df-bc 13517 df-hash 13545 df-shft 14264 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-limsup 14666 df-clim 14683 df-rlim 14684 df-sum 14881 df-ef 15258 df-sin 15260 df-cos 15261 |
This theorem is referenced by: cos2t 15368 cos2tsin 15369 sinbnd 15370 cosbnd 15371 absefi 15386 sinhalfpilem 24736 sincos6thpi 24788 efif1olem4 24814 heron 25101 asinsin 25155 atandmtan 25183 basellem8 25351 sin2h 34434 tan2h 34436 dvtan 34494 itgsinexp 41803 onetansqsecsq 44362 cotsqcscsq 44363 |
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