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Theorem cotsqcscsq 28232
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 28219 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
21oveq1d 5873 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( cot `  A
) ^ 2 )  =  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 ) )
32oveq2d 5874 . 2  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( cot `  A
) ^ 2 ) )  =  ( 1  +  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 ) ) )
4 sincossq 12456 . . . . 5  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
54oveq1d 5873 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) ) )
65adantr 451 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  / 
( ( sin `  A
) ^ 2 ) ) )
7 sincl 12406 . . . . . . . 8  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
87sqcld 11243 . . . . . . 7  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
98adantr 451 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  e.  CC )
10 sqne0 11170 . . . . . . . 8  |-  ( ( sin `  A )  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  =/=  0  <->  ( sin `  A )  =/=  0
) )
117, 10syl 15 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  =/=  0  <->  ( sin `  A )  =/=  0
) )
1211biimpar 471 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  =/=  0 )
139, 12dividd 9534 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) )  =  1 )
1413oveq1d 5873 . . . 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 12407 . . . . . . 7  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
1615sqcld 11243 . . . . . 6  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
1716adantr 451 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  e.  CC )
189, 17, 9, 12divdird 9574 . . . 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 518 . . . . . 6  |-  ( A  e.  CC  ->  (
( cos `  A
)  e.  CC  /\  ( sin `  A )  e.  CC ) )
20 2nn0 9982 . . . . . . . 8  |-  2  e.  NN0
21 expdiv 11152 . . . . . . . 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 1266 . . . . . . 7  |-  ( ( ( cos `  A
)  e.  CC  /\  ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 ) )  ->  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 )  =  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2322anassrs 629 . . . . . 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 457 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 )  =  ( ( ( cos `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2524oveq2d 5874 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( ( cos `  A
)  /  ( sin `  A ) ) ^
2 ) )  =  ( 1  +  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) ) )
2614, 18, 253eqtr4rd 2326 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( ( cos `  A
)  /  ( sin `  A ) ) ^
2 ) )  =  ( ( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) ) )
27 cscval 28218 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )
2827oveq1d 5873 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( ( 1  /  ( sin `  A
) ) ^ 2 ) )
29 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
30 expdiv 11152 . . . . . . 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 1268 . . . . . 6  |-  ( ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
327, 31sylan 457 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
33 sq1 11198 . . . . . 6  |-  ( 1 ^ 2 )  =  1
3433oveq1i 5868 . . . . 5  |-  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) )
3532, 34syl6eq 2331 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) ) )
3628, 35eqtrd 2315 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( 1  / 
( ( sin `  A
) ^ 2 ) ) )
376, 26, 363eqtr4rd 2326 . 2  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( 1  +  ( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 ) ) )
383, 37eqtr4d 2318 1  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( cot `  A
) ^ 2 ) )  =  ( ( csc `  A ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    / cdiv 9423   2c2 9795   NN0cn0 9965   ^cexp 11104   sincsin 12345   cosccos 12346   cscccsc 28212   cotccot 28213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-csc 28215  df-cot 28216
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