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Theorem onetansqsecsq 28222
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 12687 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
2 sqeq0 11405 . . . . . . . . . 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 2606 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  =/=  0  <->  ( cos `  A )  =/=  0
) )
54biimpar 472 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  =/=  0 )
61sqcld 11480 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
7 divid 9665 . . . . . . . 8  |-  ( ( ( ( cos `  A
) ^ 2 )  e.  CC  /\  (
( cos `  A
) ^ 2 )  =/=  0 )  -> 
( ( ( cos `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  1 )
86, 7sylan 458 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( ( cos `  A
) ^ 2 )  =/=  0 )  -> 
( ( ( cos `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  1 )
95, 8syldan 457 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( cos `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  1 )
109eqcomd 2413 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
1  =  ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
11 tanval 12688 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( tan `  A
)  =  ( ( sin `  A )  /  ( cos `  A
) ) )
1211oveq1d 6059 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A )  /  ( cos `  A
) ) ^ 2 ) )
13 2nn0 10198 . . . . . . . . . 10  |-  2  e.  NN0
14 sincl 12686 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
15 expdiv 11389 . . . . . . . . . . 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 1217 . . . . . . . . . 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 1268 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ( ( cos `  A
)  e.  CC  /\  ( cos `  A )  =/=  0 ) )  ->  ( ( ( sin `  A )  /  ( cos `  A
) ) ^ 2 )  =  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
18173impb 1149 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( cos `  A )  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  (
( ( sin `  A
)  /  ( cos `  A ) ) ^
2 )  =  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
191, 18syl3an2 1218 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  (
( ( sin `  A
)  /  ( cos `  A ) ) ^
2 )  =  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
20193anidm12 1241 . . . . . 6  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( sin `  A )  /  ( cos `  A ) ) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
2112, 20eqtrd 2440 . . . . 5  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( tan `  A
) ^ 2 )  =  ( ( ( sin `  A ) ^ 2 )  / 
( ( cos `  A
) ^ 2 ) ) )
2210, 21oveq12d 6062 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  +  ( ( ( sin `  A
) ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) ) )
2314sqcld 11480 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
24 divdir 9661 . . . . . . . . . . 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 1217 . . . . . . . . . 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 1218 . . . . . . . . 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 1241 . . . . . . . 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 1149 . . . . . . 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 1218 . . . . . 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 1241 . . . . 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 457 . . . 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 2443 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  (
( cos `  A
) ^ 2 ) ) )
3323, 6addcomd 9228 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  ( ( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) ) )
34 sincossq 12736 . . . . . 6  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
3533, 34eqtr3d 2442 . . . . 5  |-  ( A  e.  CC  ->  (
( ( cos `  A
) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  =  1 )
3635oveq1d 6059 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A ) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) ) )
3736adantr 452 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( ( ( cos `  A ) ^ 2 )  +  ( ( sin `  A
) ^ 2 ) )  /  ( ( cos `  A ) ^ 2 ) )  =  ( 1  / 
( ( cos `  A
) ^ 2 ) ) )
3832, 37eqtrd 2440 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( 1  +  ( ( tan `  A
) ^ 2 ) )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) ) )
39 secval 28208 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( sec `  A
)  =  ( 1  /  ( cos `  A
) ) )
4039oveq1d 6059 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sec `  A
) ^ 2 )  =  ( ( 1  /  ( cos `  A
) ) ^ 2 ) )
41 ax-1cn 9008 . . . . . 6  |-  1  e.  CC
42 expdiv 11389 . . . . . 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 1270 . . . . 5  |-  ( ( ( cos `  A
)  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( 1  / 
( cos `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
441, 43sylan 458 . . . 4  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( 1  / 
( cos `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( cos `  A ) ^ 2 ) ) )
45 sq1 11435 . . . . 5  |-  ( 1 ^ 2 )  =  1
4645oveq1i 6054 . . . 4  |-  ( ( 1 ^ 2 )  /  ( ( cos `  A ) ^ 2 ) )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) )
4744, 46syl6eq 2456 . . 3  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( 1  / 
( cos `  A
) ) ^ 2 )  =  ( 1  /  ( ( cos `  A ) ^ 2 ) ) )
4840, 47eqtrd 2440 . 2  |-  ( ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  -> 
( ( sec `  A
) ^ 2 )  =  ( 1  / 
( ( cos `  A
) ^ 2 ) ) )
4938, 48eqtr4d 2443 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   ` cfv 5417  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951    + caddc 8953    / cdiv 9637   2c2 10009   NN0cn0 10181   ^cexp 11341   sincsin 12625   cosccos 12626   tanctan 12627   seccsec 28202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029  ax-mulf 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-ico 10882  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-fac 11526  df-bc 11553  df-hash 11578  df-shft 11841  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-limsup 12224  df-clim 12241  df-rlim 12242  df-sum 12439  df-ef 12629  df-sin 12631  df-cos 12632  df-tan 12633  df-sec 28205
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