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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 | β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 11500 | . . 3 β’ (π΄ β β β -π΄ β β) | |
2 | cosadd 16151 | . . 3 β’ ((π΄ β β β§ -π΄ β β) β (cosβ(π΄ + -π΄)) = (((cosβπ΄) Β· (cosβ-π΄)) β ((sinβπ΄) Β· (sinβ-π΄)))) | |
3 | 1, 2 | mpdan 685 | . 2 β’ (π΄ β β β (cosβ(π΄ + -π΄)) = (((cosβπ΄) Β· (cosβ-π΄)) β ((sinβπ΄) Β· (sinβ-π΄)))) |
4 | negid 11547 | . . . 4 β’ (π΄ β β β (π΄ + -π΄) = 0) | |
5 | 4 | fveq2d 6906 | . . 3 β’ (π΄ β β β (cosβ(π΄ + -π΄)) = (cosβ0)) |
6 | cos0 16136 | . . 3 β’ (cosβ0) = 1 | |
7 | 5, 6 | eqtrdi 2784 | . 2 β’ (π΄ β β β (cosβ(π΄ + -π΄)) = 1) |
8 | sincl 16112 | . . . . 5 β’ (π΄ β β β (sinβπ΄) β β) | |
9 | 8 | sqcld 14150 | . . . 4 β’ (π΄ β β β ((sinβπ΄)β2) β β) |
10 | coscl 16113 | . . . . 5 β’ (π΄ β β β (cosβπ΄) β β) | |
11 | 10 | sqcld 14150 | . . . 4 β’ (π΄ β β β ((cosβπ΄)β2) β β) |
12 | 9, 11 | addcomd 11456 | . . 3 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = (((cosβπ΄)β2) + ((sinβπ΄)β2))) |
13 | 10 | sqvald 14149 | . . . . 5 β’ (π΄ β β β ((cosβπ΄)β2) = ((cosβπ΄) Β· (cosβπ΄))) |
14 | cosneg 16133 | . . . . . 6 β’ (π΄ β β β (cosβ-π΄) = (cosβπ΄)) | |
15 | 14 | oveq2d 7442 | . . . . 5 β’ (π΄ β β β ((cosβπ΄) Β· (cosβ-π΄)) = ((cosβπ΄) Β· (cosβπ΄))) |
16 | 13, 15 | eqtr4d 2771 | . . . 4 β’ (π΄ β β β ((cosβπ΄)β2) = ((cosβπ΄) Β· (cosβ-π΄))) |
17 | 8 | sqvald 14149 | . . . . . 6 β’ (π΄ β β β ((sinβπ΄)β2) = ((sinβπ΄) Β· (sinβπ΄))) |
18 | sinneg 16132 | . . . . . . . . 9 β’ (π΄ β β β (sinβ-π΄) = -(sinβπ΄)) | |
19 | 18 | negeqd 11494 | . . . . . . . 8 β’ (π΄ β β β -(sinβ-π΄) = --(sinβπ΄)) |
20 | 8 | negnegd 11602 | . . . . . . . 8 β’ (π΄ β β β --(sinβπ΄) = (sinβπ΄)) |
21 | 19, 20 | eqtrd 2768 | . . . . . . 7 β’ (π΄ β β β -(sinβ-π΄) = (sinβπ΄)) |
22 | 21 | oveq2d 7442 | . . . . . 6 β’ (π΄ β β β ((sinβπ΄) Β· -(sinβ-π΄)) = ((sinβπ΄) Β· (sinβπ΄))) |
23 | 17, 22 | eqtr4d 2771 | . . . . 5 β’ (π΄ β β β ((sinβπ΄)β2) = ((sinβπ΄) Β· -(sinβ-π΄))) |
24 | 1 | sincld 16116 | . . . . . 6 β’ (π΄ β β β (sinβ-π΄) β β) |
25 | 8, 24 | mulneg2d 11708 | . . . . 5 β’ (π΄ β β β ((sinβπ΄) Β· -(sinβ-π΄)) = -((sinβπ΄) Β· (sinβ-π΄))) |
26 | 23, 25 | eqtrd 2768 | . . . 4 β’ (π΄ β β β ((sinβπ΄)β2) = -((sinβπ΄) Β· (sinβ-π΄))) |
27 | 16, 26 | oveq12d 7444 | . . 3 β’ (π΄ β β β (((cosβπ΄)β2) + ((sinβπ΄)β2)) = (((cosβπ΄) Β· (cosβ-π΄)) + -((sinβπ΄) Β· (sinβ-π΄)))) |
28 | 1 | coscld 16117 | . . . . 5 β’ (π΄ β β β (cosβ-π΄) β β) |
29 | 10, 28 | mulcld 11274 | . . . 4 β’ (π΄ β β β ((cosβπ΄) Β· (cosβ-π΄)) β β) |
30 | 8, 24 | mulcld 11274 | . . . 4 β’ (π΄ β β β ((sinβπ΄) Β· (sinβ-π΄)) β β) |
31 | 29, 30 | negsubd 11617 | . . 3 β’ (π΄ β β β (((cosβπ΄) Β· (cosβ-π΄)) + -((sinβπ΄) Β· (sinβ-π΄))) = (((cosβπ΄) Β· (cosβ-π΄)) β ((sinβπ΄) Β· (sinβ-π΄)))) |
32 | 12, 27, 31 | 3eqtrrd 2773 | . 2 β’ (π΄ β β β (((cosβπ΄) Β· (cosβ-π΄)) β ((sinβπ΄) Β· (sinβ-π΄))) = (((sinβπ΄)β2) + ((cosβπ΄)β2))) |
33 | 3, 7, 32 | 3eqtr3rd 2777 | 1 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βcc 11146 0cc0 11148 1c1 11149 + caddc 11151 Β· cmul 11153 β cmin 11484 -cneg 11485 2c2 12307 βcexp 14068 sincsin 16049 cosccos 16050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-ico 13372 df-fz 13527 df-fzo 13670 df-fl 13799 df-seq 14009 df-exp 14069 df-fac 14275 df-bc 14304 df-hash 14332 df-shft 15056 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-ef 16053 df-sin 16055 df-cos 16056 |
This theorem is referenced by: cos2t 16164 cos2tsin 16165 sinbnd 16166 cosbnd 16167 absefi 16182 sinhalfpilem 26426 sincos6thpi 26478 efif1olem4 26507 heron 26798 asinsin 26852 atandmtan 26880 basellem8 27048 sin2h 37124 tan2h 37126 dvtan 37184 itgsinexp 45390 onetansqsecsq 48288 cotsqcscsq 48289 |
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