Theorem cotsqcscsq 42503

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

Step	Hyp	Ref	Expression
1		cotval 42490	. . . 4 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( cot ` A ) = ( ( cos ` A ) / ( sin ` A ) ) )
2	1	oveq1d 6665	. . 3 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( cot ` A ) ^ 2 ) = ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) )
3	2	oveq2d 6666	. 2 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( cot ` A ) ^ 2 ) ) = ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) )
4		sincossq 14906	. . . . 5 |- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) = 1 )
5	4	oveq1d 6665	. . . 4 |- ( A e. CC -> ( ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) / ( ( sin ` A ) ^ 2 ) ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) )
6	5	adantr 481	. . 3 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) / ( ( sin ` A ) ^ 2 ) ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) )
7		sincl 14856	. . . . . . . 8 |- ( A e. CC -> ( sin ` A ) e. CC )
8	7	sqcld 13006	. . . . . . 7 |- ( A e. CC -> ( ( sin ` A ) ^ 2 ) e. CC )
9	8	adantr 481	. . . . . 6 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( sin ` A ) ^ 2 ) e. CC )
10		sqne0 12930	. . . . . . . 8 |- ( ( sin ` A ) e. CC -> ( ( ( sin ` A ) ^ 2 ) =/= 0 <-> ( sin ` A ) =/= 0 ) )
11	7, 10	syl 17	. . . . . . 7 |- ( A e. CC -> ( ( ( sin ` A ) ^ 2 ) =/= 0 <-> ( sin ` A ) =/= 0 ) )
12	11	biimpar 502	. . . . . 6 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( sin ` A ) ^ 2 ) =/= 0 )
13	9, 12	dividd 10799	. . . . 5 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( sin ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) = 1 )
14	13	oveq1d 6665	. . . 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 14857	. . . . . . 7 |- ( A e. CC -> ( cos ` A ) e. CC )
16	15	sqcld 13006	. . . . . 6 |- ( A e. CC -> ( ( cos ` A ) ^ 2 ) e. CC )
17	16	adantr 481	. . . . 5 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( cos ` A ) ^ 2 ) e. CC )
18	9, 17, 9, 12	divdird 10839	. . . 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 ) ) ) )
19	15, 7	jca 554	. . . . . 6 |- ( A e. CC -> ( ( cos ` A ) e. CC /\ ( sin ` A ) e. CC ) )
20		2nn0 11309	. . . . . . . 8 |- 2 e. NN0
21		expdiv 12911	. . . . . . . 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 ) ) )
22	20, 21	mp3an3 1413	. . . . . . 7 |- ( ( ( cos ` A ) e. CC /\ ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) ) -> ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) = ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) )
23	22	anassrs 680	. . . . . 6 |- ( ( ( ( cos ` A ) e. CC /\ ( sin ` A ) e. CC ) /\ ( sin ` A ) =/= 0 ) -> ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) = ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) )
24	19, 23	sylan 488	. . . . 5 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) = ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) )
25	24	oveq2d 6666	. . . 4 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) = ( 1 + ( ( ( cos ` A ) ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) ) )
26	14, 18, 25	3eqtr4rd 2667	. . 3 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) = ( ( ( ( sin ` A ) ^ 2 ) + ( ( cos ` A ) ^ 2 ) ) / ( ( sin ` A ) ^ 2 ) ) )
27		cscval 42489	. . . . 5 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( csc ` A ) = ( 1 / ( sin ` A ) ) )
28	27	oveq1d 6665	. . . 4 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( csc ` A ) ^ 2 ) = ( ( 1 / ( sin ` A ) ) ^ 2 ) )
29		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
30		expdiv 12911	. . . . . . 7 |- ( ( 1 e. CC /\ ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) /\ 2 e. NN0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) )
31	29, 20, 30	mp3an13 1415	. . . . . 6 |- ( ( ( sin ` A ) e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) )
32	7, 31	sylan 488	. . . . 5 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) )
33		sq1 12958	. . . . . 6 |- ( 1 ^ 2 ) = 1
34	33	oveq1i 6660	. . . . 5 |- ( ( 1 ^ 2 ) / ( ( sin ` A ) ^ 2 ) ) = ( 1 / ( ( sin ` A ) ^ 2 ) )
35	32, 34	syl6eq 2672	. . . 4 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( 1 / ( sin ` A ) ) ^ 2 ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) )
36	28, 35	eqtrd 2656	. . 3 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( csc ` A ) ^ 2 ) = ( 1 / ( ( sin ` A ) ^ 2 ) ) )
37	6, 26, 36	3eqtr4rd 2667	. 2 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( ( csc ` A ) ^ 2 ) = ( 1 + ( ( ( cos ` A ) / ( sin ` A ) ) ^ 2 ) ) )
38	3, 37	eqtr4d 2659	1 |- ( ( A e. CC /\ ( sin ` A ) =/= 0 ) -> ( 1 + ( ( cot ` A ) ^ 2 ) ) = ( ( csc ` A ) ^ 2 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   / cdiv 10684   2c2 11070   NN0cn0 11292   ^cexp 12860   sincsin 14794   cosccos 14795   cscccsc 42483   cotccot 42484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-csc 42486  df-cot 42487

This theorem is referenced by: (None)