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Mirrors > Home > ILE Home > Th. List > tannegap | GIF version |
Description: The tangent of a negative is the negative of the tangent. (Contributed by David A. Wheeler, 23-Mar-2014.) |
Ref | Expression |
---|---|
tannegap | β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβ-π΄) = -(tanβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coscl 11734 | . . . 4 β’ (π΄ β β β (cosβπ΄) β β) | |
2 | sincl 11733 | . . . . 5 β’ (π΄ β β β (sinβπ΄) β β) | |
3 | divnegap 8682 | . . . . 5 β’ (((sinβπ΄) β β β§ (cosβπ΄) β β β§ (cosβπ΄) # 0) β -((sinβπ΄) / (cosβπ΄)) = (-(sinβπ΄) / (cosβπ΄))) | |
4 | 2, 3 | syl3an1 1282 | . . . 4 β’ ((π΄ β β β§ (cosβπ΄) β β β§ (cosβπ΄) # 0) β -((sinβπ΄) / (cosβπ΄)) = (-(sinβπ΄) / (cosβπ΄))) |
5 | 1, 4 | syl3an2 1283 | . . 3 β’ ((π΄ β β β§ π΄ β β β§ (cosβπ΄) # 0) β -((sinβπ΄) / (cosβπ΄)) = (-(sinβπ΄) / (cosβπ΄))) |
6 | 5 | 3anidm12 1306 | . 2 β’ ((π΄ β β β§ (cosβπ΄) # 0) β -((sinβπ΄) / (cosβπ΄)) = (-(sinβπ΄) / (cosβπ΄))) |
7 | tanvalap 11735 | . . 3 β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβπ΄) = ((sinβπ΄) / (cosβπ΄))) | |
8 | 7 | negeqd 8171 | . 2 β’ ((π΄ β β β§ (cosβπ΄) # 0) β -(tanβπ΄) = -((sinβπ΄) / (cosβπ΄))) |
9 | negcl 8176 | . . . 4 β’ (π΄ β β β -π΄ β β) | |
10 | cosneg 11754 | . . . . . 6 β’ (π΄ β β β (cosβ-π΄) = (cosβπ΄)) | |
11 | 10 | adantr 276 | . . . . 5 β’ ((π΄ β β β§ (cosβπ΄) # 0) β (cosβ-π΄) = (cosβπ΄)) |
12 | simpr 110 | . . . . 5 β’ ((π΄ β β β§ (cosβπ΄) # 0) β (cosβπ΄) # 0) | |
13 | 11, 12 | eqbrtrd 4040 | . . . 4 β’ ((π΄ β β β§ (cosβπ΄) # 0) β (cosβ-π΄) # 0) |
14 | tanvalap 11735 | . . . 4 β’ ((-π΄ β β β§ (cosβ-π΄) # 0) β (tanβ-π΄) = ((sinβ-π΄) / (cosβ-π΄))) | |
15 | 9, 13, 14 | syl2an2r 595 | . . 3 β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβ-π΄) = ((sinβ-π΄) / (cosβ-π΄))) |
16 | sinneg 11753 | . . . . 5 β’ (π΄ β β β (sinβ-π΄) = -(sinβπ΄)) | |
17 | 16, 10 | oveq12d 5909 | . . . 4 β’ (π΄ β β β ((sinβ-π΄) / (cosβ-π΄)) = (-(sinβπ΄) / (cosβπ΄))) |
18 | 17 | adantr 276 | . . 3 β’ ((π΄ β β β§ (cosβπ΄) # 0) β ((sinβ-π΄) / (cosβ-π΄)) = (-(sinβπ΄) / (cosβπ΄))) |
19 | 15, 18 | eqtrd 2222 | . 2 β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβ-π΄) = (-(sinβπ΄) / (cosβπ΄))) |
20 | 6, 8, 19 | 3eqtr4rd 2233 | 1 β’ ((π΄ β β β§ (cosβπ΄) # 0) β (tanβ-π΄) = -(tanβπ΄)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1364 β wcel 2160 class class class wbr 4018 βcfv 5231 (class class class)co 5891 βcc 7828 0cc0 7830 -cneg 8148 # cap 8557 / cdiv 8648 sincsin 11671 cosccos 11672 tanctan 11673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 ax-caucvg 7950 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-en 6759 df-dom 6760 df-fin 6761 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-n0 9196 df-z 9273 df-uz 9548 df-q 9639 df-rp 9673 df-ico 9913 df-fz 10028 df-fzo 10162 df-seqfrec 10465 df-exp 10539 df-fac 10725 df-ihash 10775 df-cj 10870 df-re 10871 df-im 10872 df-rsqrt 11026 df-abs 11027 df-clim 11306 df-sumdc 11381 df-ef 11675 df-sin 11677 df-cos 11678 df-tan 11679 |
This theorem is referenced by: (None) |
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