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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 11406 | . . 3 β’ (π΄ β β β -π΄ β β) | |
2 | cosadd 16052 | . . 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 686 | . 2 β’ (π΄ β β β (cosβ(π΄ + -π΄)) = (((cosβπ΄) Β· (cosβ-π΄)) β ((sinβπ΄) Β· (sinβ-π΄)))) |
4 | negid 11453 | . . . 4 β’ (π΄ β β β (π΄ + -π΄) = 0) | |
5 | 4 | fveq2d 6847 | . . 3 β’ (π΄ β β β (cosβ(π΄ + -π΄)) = (cosβ0)) |
6 | cos0 16037 | . . 3 β’ (cosβ0) = 1 | |
7 | 5, 6 | eqtrdi 2789 | . 2 β’ (π΄ β β β (cosβ(π΄ + -π΄)) = 1) |
8 | sincl 16013 | . . . . 5 β’ (π΄ β β β (sinβπ΄) β β) | |
9 | 8 | sqcld 14055 | . . . 4 β’ (π΄ β β β ((sinβπ΄)β2) β β) |
10 | coscl 16014 | . . . . 5 β’ (π΄ β β β (cosβπ΄) β β) | |
11 | 10 | sqcld 14055 | . . . 4 β’ (π΄ β β β ((cosβπ΄)β2) β β) |
12 | 9, 11 | addcomd 11362 | . . 3 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = (((cosβπ΄)β2) + ((sinβπ΄)β2))) |
13 | 10 | sqvald 14054 | . . . . 5 β’ (π΄ β β β ((cosβπ΄)β2) = ((cosβπ΄) Β· (cosβπ΄))) |
14 | cosneg 16034 | . . . . . 6 β’ (π΄ β β β (cosβ-π΄) = (cosβπ΄)) | |
15 | 14 | oveq2d 7374 | . . . . 5 β’ (π΄ β β β ((cosβπ΄) Β· (cosβ-π΄)) = ((cosβπ΄) Β· (cosβπ΄))) |
16 | 13, 15 | eqtr4d 2776 | . . . 4 β’ (π΄ β β β ((cosβπ΄)β2) = ((cosβπ΄) Β· (cosβ-π΄))) |
17 | 8 | sqvald 14054 | . . . . . 6 β’ (π΄ β β β ((sinβπ΄)β2) = ((sinβπ΄) Β· (sinβπ΄))) |
18 | sinneg 16033 | . . . . . . . . 9 β’ (π΄ β β β (sinβ-π΄) = -(sinβπ΄)) | |
19 | 18 | negeqd 11400 | . . . . . . . 8 β’ (π΄ β β β -(sinβ-π΄) = --(sinβπ΄)) |
20 | 8 | negnegd 11508 | . . . . . . . 8 β’ (π΄ β β β --(sinβπ΄) = (sinβπ΄)) |
21 | 19, 20 | eqtrd 2773 | . . . . . . 7 β’ (π΄ β β β -(sinβ-π΄) = (sinβπ΄)) |
22 | 21 | oveq2d 7374 | . . . . . 6 β’ (π΄ β β β ((sinβπ΄) Β· -(sinβ-π΄)) = ((sinβπ΄) Β· (sinβπ΄))) |
23 | 17, 22 | eqtr4d 2776 | . . . . 5 β’ (π΄ β β β ((sinβπ΄)β2) = ((sinβπ΄) Β· -(sinβ-π΄))) |
24 | 1 | sincld 16017 | . . . . . 6 β’ (π΄ β β β (sinβ-π΄) β β) |
25 | 8, 24 | mulneg2d 11614 | . . . . 5 β’ (π΄ β β β ((sinβπ΄) Β· -(sinβ-π΄)) = -((sinβπ΄) Β· (sinβ-π΄))) |
26 | 23, 25 | eqtrd 2773 | . . . 4 β’ (π΄ β β β ((sinβπ΄)β2) = -((sinβπ΄) Β· (sinβ-π΄))) |
27 | 16, 26 | oveq12d 7376 | . . 3 β’ (π΄ β β β (((cosβπ΄)β2) + ((sinβπ΄)β2)) = (((cosβπ΄) Β· (cosβ-π΄)) + -((sinβπ΄) Β· (sinβ-π΄)))) |
28 | 1 | coscld 16018 | . . . . 5 β’ (π΄ β β β (cosβ-π΄) β β) |
29 | 10, 28 | mulcld 11180 | . . . 4 β’ (π΄ β β β ((cosβπ΄) Β· (cosβ-π΄)) β β) |
30 | 8, 24 | mulcld 11180 | . . . 4 β’ (π΄ β β β ((sinβπ΄) Β· (sinβ-π΄)) β β) |
31 | 29, 30 | negsubd 11523 | . . 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 2778 | . 2 β’ (π΄ β β β (((cosβπ΄) Β· (cosβ-π΄)) β ((sinβπ΄) Β· (sinβ-π΄))) = (((sinβπ΄)β2) + ((cosβπ΄)β2))) |
33 | 3, 7, 32 | 3eqtr3rd 2782 | 1 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 βcc 11054 0cc0 11056 1c1 11057 + caddc 11059 Β· cmul 11061 β cmin 11390 -cneg 11391 2c2 12213 βcexp 13973 sincsin 15951 cosccos 15952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-ico 13276 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-fac 14180 df-bc 14209 df-hash 14237 df-shft 14958 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 df-rlim 15377 df-sum 15577 df-ef 15955 df-sin 15957 df-cos 15958 |
This theorem is referenced by: cos2t 16065 cos2tsin 16066 sinbnd 16067 cosbnd 16068 absefi 16083 sinhalfpilem 25836 sincos6thpi 25888 efif1olem4 25917 heron 26204 asinsin 26258 atandmtan 26286 basellem8 26453 sin2h 36114 tan2h 36116 dvtan 36174 itgsinexp 44282 onetansqsecsq 47292 cotsqcscsq 47293 |
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