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Theorem cotsqcscsq 40535
Description: Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
Assertion
Ref Expression
cotsqcscsq  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( cot `  A
) ^ 2 ) )  =  ( ( csc `  A ) ^ 2 ) )

Proof of Theorem cotsqcscsq
StepHypRef Expression
1 cotval 40522 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
21oveq1d 6305 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( cot `  A
) ^ 2 )  =  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 ) )
32oveq2d 6306 . 2  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( cot `  A
) ^ 2 ) )  =  ( 1  +  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 ) ) )
4 sincossq 14230 . . . . 5  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
54oveq1d 6305 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) ) )
65adantr 467 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  / 
( ( sin `  A
) ^ 2 ) ) )
7 sincl 14180 . . . . . . . 8  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
87sqcld 12414 . . . . . . 7  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
98adantr 467 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  e.  CC )
10 sqne0 12341 . . . . . . . 8  |-  ( ( sin `  A )  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  =/=  0  <->  ( sin `  A )  =/=  0
) )
117, 10syl 17 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  =/=  0  <->  ( sin `  A )  =/=  0
) )
1211biimpar 488 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  =/=  0 )
139, 12dividd 10381 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) )  =  1 )
1413oveq1d 6305 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( ( sin `  A ) ^ 2 )  / 
( ( sin `  A
) ^ 2 ) )  +  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )  =  ( 1  +  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) ) )
15 coscl 14181 . . . . . . 7  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
1615sqcld 12414 . . . . . 6  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
1716adantr 467 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  e.  CC )
189, 17, 9, 12divdird 10421 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) )  =  ( ( ( ( sin `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) )  +  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) ) )
1915, 7jca 535 . . . . . 6  |-  ( A  e.  CC  ->  (
( cos `  A
)  e.  CC  /\  ( sin `  A )  e.  CC ) )
20 2nn0 10886 . . . . . . . 8  |-  2  e.  NN0
21 expdiv 12323 . . . . . . . 8  |-  ( ( ( cos `  A
)  e.  CC  /\  ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 )  /\  2  e.  NN0 )  -> 
( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 )  =  ( ( ( cos `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2220, 21mp3an3 1353 . . . . . . 7  |-  ( ( ( cos `  A
)  e.  CC  /\  ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 ) )  ->  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 )  =  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2322anassrs 654 . . . . . 6  |-  ( ( ( ( cos `  A
)  e.  CC  /\  ( sin `  A )  e.  CC )  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 )  =  ( ( ( cos `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2419, 23sylan 474 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 )  =  ( ( ( cos `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2524oveq2d 6306 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( ( cos `  A
)  /  ( sin `  A ) ) ^
2 ) )  =  ( 1  +  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) ) )
2614, 18, 253eqtr4rd 2496 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( ( cos `  A
)  /  ( sin `  A ) ) ^
2 ) )  =  ( ( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) ) )
27 cscval 40521 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )
2827oveq1d 6305 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( ( 1  /  ( sin `  A
) ) ^ 2 ) )
29 ax-1cn 9597 . . . . . . 7  |-  1  e.  CC
30 expdiv 12323 . . . . . . 7  |-  ( ( 1  e.  CC  /\  ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 )  /\  2  e.  NN0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
3129, 20, 30mp3an13 1355 . . . . . 6  |-  ( ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
327, 31sylan 474 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
33 sq1 12369 . . . . . 6  |-  ( 1 ^ 2 )  =  1
3433oveq1i 6300 . . . . 5  |-  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) )
3532, 34syl6eq 2501 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) ) )
3628, 35eqtrd 2485 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( 1  / 
( ( sin `  A
) ^ 2 ) ) )
376, 26, 363eqtr4rd 2496 . 2  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( 1  +  ( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 ) ) )
383, 37eqtr4d 2488 1  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( cot `  A
) ^ 2 ) )  =  ( ( csc `  A ) ^ 2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   ` cfv 5582  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    / cdiv 10269   2c2 10659   NN0cn0 10869   ^cexp 12272   sincsin 14116   cosccos 14117   cscccsc 40515   cotccot 40516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-csc 40518  df-cot 40519
This theorem is referenced by: (None)
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