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Mirrors > Home > MPE Home > Th. List > cos2t | Structured version Visualization version GIF version |
Description: Double-angle formula for cosine. (Contributed by Paul Chapman, 24-Jan-2008.) |
Ref | Expression |
---|---|
cos2t | ⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coscl 16008 | . . . 4 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
2 | 1 | sqcld 14048 | . . 3 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) ∈ ℂ) |
3 | ax-1cn 11108 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | subsub3 11432 | . . . 4 ⊢ ((((cos‘𝐴)↑2) ∈ ℂ ∧ 1 ∈ ℂ ∧ ((cos‘𝐴)↑2) ∈ ℂ) → (((cos‘𝐴)↑2) − (1 − ((cos‘𝐴)↑2))) = ((((cos‘𝐴)↑2) + ((cos‘𝐴)↑2)) − 1)) | |
5 | 3, 4 | mp3an2 1449 | . . 3 ⊢ ((((cos‘𝐴)↑2) ∈ ℂ ∧ ((cos‘𝐴)↑2) ∈ ℂ) → (((cos‘𝐴)↑2) − (1 − ((cos‘𝐴)↑2))) = ((((cos‘𝐴)↑2) + ((cos‘𝐴)↑2)) − 1)) |
6 | 2, 2, 5 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) − (1 − ((cos‘𝐴)↑2))) = ((((cos‘𝐴)↑2) + ((cos‘𝐴)↑2)) − 1)) |
7 | cosadd 16046 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (cos‘(𝐴 + 𝐴)) = (((cos‘𝐴) · (cos‘𝐴)) − ((sin‘𝐴) · (sin‘𝐴)))) | |
8 | 7 | anidms 567 | . . . 4 ⊢ (𝐴 ∈ ℂ → (cos‘(𝐴 + 𝐴)) = (((cos‘𝐴) · (cos‘𝐴)) − ((sin‘𝐴) · (sin‘𝐴)))) |
9 | 2times 12288 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
10 | 9 | fveq2d 6846 | . . . 4 ⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = (cos‘(𝐴 + 𝐴))) |
11 | 1 | sqvald 14047 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((cos‘𝐴)↑2) = ((cos‘𝐴) · (cos‘𝐴))) |
12 | sincl 16007 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
13 | 12 | sqvald 14047 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) = ((sin‘𝐴) · (sin‘𝐴))) |
14 | 11, 13 | oveq12d 7374 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) − ((sin‘𝐴)↑2)) = (((cos‘𝐴) · (cos‘𝐴)) − ((sin‘𝐴) · (sin‘𝐴)))) |
15 | 8, 10, 14 | 3eqtr4d 2786 | . . 3 ⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = (((cos‘𝐴)↑2) − ((sin‘𝐴)↑2))) |
16 | 12 | sqcld 14048 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((sin‘𝐴)↑2) ∈ ℂ) |
17 | 16, 2 | addcomd 11356 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2))) |
18 | sincossq 16057 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) | |
19 | 17, 18 | eqtr3d 2778 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = 1) |
20 | subadd 11403 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ ((cos‘𝐴)↑2) ∈ ℂ ∧ ((sin‘𝐴)↑2) ∈ ℂ) → ((1 − ((cos‘𝐴)↑2)) = ((sin‘𝐴)↑2) ↔ (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = 1)) | |
21 | 3, 2, 16, 20 | mp3an2i 1466 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((1 − ((cos‘𝐴)↑2)) = ((sin‘𝐴)↑2) ↔ (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = 1)) |
22 | 19, 21 | mpbird 256 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 − ((cos‘𝐴)↑2)) = ((sin‘𝐴)↑2)) |
23 | 22 | oveq2d 7372 | . . 3 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) − (1 − ((cos‘𝐴)↑2))) = (((cos‘𝐴)↑2) − ((sin‘𝐴)↑2))) |
24 | 15, 23 | eqtr4d 2779 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = (((cos‘𝐴)↑2) − (1 − ((cos‘𝐴)↑2)))) |
25 | 2 | 2timesd 12395 | . . 3 ⊢ (𝐴 ∈ ℂ → (2 · ((cos‘𝐴)↑2)) = (((cos‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
26 | 25 | oveq1d 7371 | . 2 ⊢ (𝐴 ∈ ℂ → ((2 · ((cos‘𝐴)↑2)) − 1) = ((((cos‘𝐴)↑2) + ((cos‘𝐴)↑2)) − 1)) |
27 | 6, 24, 26 | 3eqtr4d 2786 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘(2 · 𝐴)) = ((2 · ((cos‘𝐴)↑2)) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7356 ℂcc 11048 1c1 11051 + caddc 11053 · cmul 11055 − cmin 11384 2c2 12207 ↑cexp 13966 sincsin 15945 cosccos 15946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-z 12499 df-uz 12763 df-rp 12915 df-ico 13269 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14951 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-limsup 15352 df-clim 15369 df-rlim 15370 df-sum 15570 df-ef 15949 df-sin 15951 df-cos 15952 |
This theorem is referenced by: cos2tsin 16060 cos2bnd 16069 cospi 25827 cos2pi 25831 tangtx 25860 coskpi 25877 sin2h 36059 cos2h 36060 |
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