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Theorem cotsqcscsq 32866
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 32853 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( cot `  A
)  =  ( ( cos `  A )  /  ( sin `  A
) ) )
21oveq1d 6292 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( cot `  A
) ^ 2 )  =  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 ) )
32oveq2d 6293 . 2  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( cot `  A
) ^ 2 ) )  =  ( 1  +  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 ) ) )
4 sincossq 13783 . . . . 5  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  =  1 )
54oveq1d 6292 . . . 4  |-  ( A  e.  CC  ->  (
( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A ) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) ) )
65adantr 465 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  / 
( ( sin `  A
) ^ 2 ) ) )
7 sincl 13733 . . . . . . . 8  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
87sqcld 12282 . . . . . . 7  |-  ( A  e.  CC  ->  (
( sin `  A
) ^ 2 )  e.  CC )
98adantr 465 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  e.  CC )
10 sqne0 12208 . . . . . . . 8  |-  ( ( sin `  A )  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  =/=  0  <->  ( sin `  A )  =/=  0
) )
117, 10syl 16 . . . . . . 7  |-  ( A  e.  CC  ->  (
( ( sin `  A
) ^ 2 )  =/=  0  <->  ( sin `  A )  =/=  0
) )
1211biimpar 485 . . . . . 6  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( sin `  A
) ^ 2 )  =/=  0 )
139, 12dividd 10319 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( sin `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) )  =  1 )
1413oveq1d 6292 . . . 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 13734 . . . . . . 7  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
1615sqcld 12282 . . . . . 6  |-  ( A  e.  CC  ->  (
( cos `  A
) ^ 2 )  e.  CC )
1716adantr 465 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( cos `  A
) ^ 2 )  e.  CC )
189, 17, 9, 12divdird 10359 . . . 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 532 . . . . . 6  |-  ( A  e.  CC  ->  (
( cos `  A
)  e.  CC  /\  ( sin `  A )  e.  CC ) )
20 2nn0 10813 . . . . . . . 8  |-  2  e.  NN0
21 expdiv 12190 . . . . . . . 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 1312 . . . . . . 7  |-  ( ( ( cos `  A
)  e.  CC  /\  ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 ) )  ->  ( ( ( cos `  A )  /  ( sin `  A
) ) ^ 2 )  =  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2322anassrs 648 . . . . . 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 471 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 )  =  ( ( ( cos `  A ) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
2524oveq2d 6293 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( ( cos `  A
)  /  ( sin `  A ) ) ^
2 ) )  =  ( 1  +  ( ( ( cos `  A
) ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) ) )
2614, 18, 253eqtr4rd 2493 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( 1  +  ( ( ( cos `  A
)  /  ( sin `  A ) ) ^
2 ) )  =  ( ( ( ( sin `  A ) ^ 2 )  +  ( ( cos `  A
) ^ 2 ) )  /  ( ( sin `  A ) ^ 2 ) ) )
27 cscval 32852 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( csc `  A
)  =  ( 1  /  ( sin `  A
) ) )
2827oveq1d 6292 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( ( 1  /  ( sin `  A
) ) ^ 2 ) )
29 ax-1cn 9548 . . . . . . 7  |-  1  e.  CC
30 expdiv 12190 . . . . . . 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 1314 . . . . . 6  |-  ( ( ( sin `  A
)  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
327, 31sylan 471 . . . . 5  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) ) )
33 sq1 12236 . . . . . 6  |-  ( 1 ^ 2 )  =  1
3433oveq1i 6287 . . . . 5  |-  ( ( 1 ^ 2 )  /  ( ( sin `  A ) ^ 2 ) )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) )
3532, 34syl6eq 2498 . . . 4  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( 1  / 
( sin `  A
) ) ^ 2 )  =  ( 1  /  ( ( sin `  A ) ^ 2 ) ) )
3628, 35eqtrd 2482 . . 3  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( 1  / 
( ( sin `  A
) ^ 2 ) ) )
376, 26, 363eqtr4rd 2493 . 2  |-  ( ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  -> 
( ( csc `  A
) ^ 2 )  =  ( 1  +  ( ( ( cos `  A )  /  ( sin `  A ) ) ^ 2 ) ) )
383, 37eqtr4d 2485 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 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   ` cfv 5574  (class class class)co 6277   CCcc 9488   0cc0 9490   1c1 9491    + caddc 9493    / cdiv 10207   2c2 10586   NN0cn0 10796   ^cexp 12140   sincsin 13672   cosccos 13673   cscccsc 32846   cotccot 32847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-ico 11539  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-fac 12328  df-bc 12355  df-hash 12380  df-shft 12874  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-limsup 13268  df-clim 13285  df-rlim 13286  df-sum 13483  df-ef 13676  df-sin 13678  df-cos 13679  df-csc 32849  df-cot 32850
This theorem is referenced by: (None)
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