Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onetansqsecsq Structured version   Unicode version

Theorem onetansqsecsq 28602
Description: Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
Assertion
Ref Expression
onetansqsecsq  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( ( sec `  A ) ^ 2 ) )

Proof of Theorem onetansqsecsq
StepHypRef Expression
1 coscl 12759 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
2 sqeq0 11477 . . . . . . . . . 10  |-  ( ( cos `  A )  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  =  0  <->  ( cos `  A )  =  0 ) )
31, 2syl 16 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  =  0  <->  ( cos `  A )  =  0 ) )
43necon3bid 2642 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  =/=  0  <->  ( cos `  A )  =/=  0
) )
54biimpar 473 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  =/=  0 )
61sqcld 11552 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
7 divid 9736 . . . . . . . 8  |-  ( ( ( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  =/=  0 )  -> 
( ( ( cos `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  1 )
86, 7sylan 459 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( cos `  A
) ^ 2 )  =/=  0 )  -> 
( ( ( cos `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  1 )
95, 8syldan 458 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( cos `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  1 )
109eqcomd 2447 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
1  =  ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
11 tanval 12760 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
1211oveq1d 6125 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A )  /  ( cos `  A
) ) ^ 2 ) )
13 2nn0 10269 . . . . . . . . . 10  |-  2  e.  NN0
14 sincl 12758 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
15 expdiv 11461 . . . . . . . . . . 11  |-  ( ( ( sin `  A
)  e.  CC  /\  ( ( cos `  A
)  e.  CC  /\  ( cos `  A )  =/=  0 )  /\  2  e.  NN0 )  -> 
( ( ( sin `  A )  /  ( cos `  A ) ) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
1614, 15syl3an1 1218 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( cos `  A
)  e.  CC  /\  ( cos `  A )  =/=  0 )  /\  2  e.  NN0 )  -> 
( ( ( sin `  A )  /  ( cos `  A ) ) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
1713, 16mp3an3 1269 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( cos `  A
)  e.  CC  /\  ( cos `  A )  =/=  0 ) )  ->  ( ( ( sin `  A )  /  ( cos `  A
) ) ^ 2 )  =  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
18173impb 1150 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  (
( ( sin `  A
)  /  ( cos `  A ) ) ^
2 )  =  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
191, 18syl3an2 1219 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  (
( ( sin `  A
)  /  ( cos `  A ) ) ^
2 )  =  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
20193anidm12 1242 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A )  /  ( cos `  A ) ) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
2112, 20eqtrd 2474 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  / 
( ( cos `  A
) ^ 2 ) ) )
2210, 21oveq12d 6128 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
2314sqcld 11552 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
24 divdir 9732 . . . . . . . . . . 11  |-  ( ( ( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( sin `  A
) ^ 2 )  e.  CC  /\  (
( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  =/=  0 ) )  ->  ( ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  (
( cos `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
256, 24syl3an1 1218 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  ( ( sin `  A
) ^ 2 )  e.  CC  /\  (
( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  =/=  0 ) )  ->  ( ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  (
( cos `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
2623, 25syl3an2 1219 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  (
( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  =/=  0 ) )  ->  ( ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  (
( cos `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
27263anidm12 1242 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( ( cos `  A ) ^ 2 )  e.  CC  /\  ( ( cos `  A
) ^ 2 )  =/=  0 ) )  ->  ( ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  (
( cos `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
28273impb 1150 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  =/=  0 )  -> 
( ( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A
) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
296, 28syl3an2 1219 . . . . . 6  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  (
( cos `  A
) ^ 2 )  =/=  0 )  -> 
( ( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A
) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
30293anidm12 1242 . . . . 5  |-  ( ( A  e.  CC  /\  ( ( cos `  A
) ^ 2 )  =/=  0 )  -> 
( ( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A
) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
315, 30syldan 458 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A
) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
3222, 31eqtr4d 2477 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  (
( cos `  A
) ^ 2 ) ) )
3323, 6addcomd 9299 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) ) )
34 sincossq 12808 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
3533, 34eqtr3d 2476 . . . . 5  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  1 )
3635oveq1d 6125 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) ) )
3736adantr 453 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A
) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( 1  / 
( ( cos `  A
) ^ 2 ) ) )
3832, 37eqtrd 2474 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) ) )
39 secval 28588 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
4039oveq1d 6125 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sec `  A
) ^ 2 )  =  ( ( 1  /  ( cos `  A
) ) ^ 2 ) )
41 ax-1cn 9079 . . . . . 6  |-  1  e.  CC
42 expdiv 11461 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( ( cos `  A
)  e.  CC  /\  ( cos `  A )  =/=  0 )  /\  2  e.  NN0 )  -> 
( ( 1  / 
( cos `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
4341, 13, 42mp3an13 1271 . . . . 5  |-  ( ( ( cos `  A
)  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( 1  / 
( cos `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
441, 43sylan 459 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( 1  / 
( cos `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
45 sq1 11507 . . . . 5  |-  ( 1 ^ 2 )  =  1
4645oveq1i 6120 . . . 4  |-  ( ( 1 ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) )
4744, 46syl6eq 2490 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( 1  / 
( cos `  A
) ) ^ 2 )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) ) )
4840, 47eqtrd 2474 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sec `  A
) ^ 2 )  =  ( 1  / 
( ( cos `  A
) ^ 2 ) ) )
4938, 48eqtr4d 2477 1  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( ( sec `  A ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   ` cfv 5483  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022    + caddc 9024    / cdiv 9708   2c2 10080   NN0cn0 10252   ^cexp 11413   sincsin 12697   cosccos 12698   tanctan 12699   seccsec 28582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-pm 7050  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-sup 7475  df-oi 7508  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-ico 10953  df-fz 11075  df-fzo 11167  df-fl 11233  df-seq 11355  df-exp 11414  df-fac 11598  df-bc 11625  df-hash 11650  df-shft 11913  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-limsup 12296  df-clim 12313  df-rlim 12314  df-sum 12511  df-ef 12701  df-sin 12703  df-cos 12704  df-tan 12705  df-sec 28585
  Copyright terms: Public domain W3C validator