
Za dokončanje dokaza bomo potrebovali ti dve identiteti:
Začel bom z desno stranjo in nato manipuliral, dokler ne bo videti kot na levi strani:
To je dokaz. Upam, da je to pomagalo!
Prizadevamo si dokazati identiteto:
(tanx + sinx) / (2tanx) - = cos ^ 2 (x / 2)
Upoštevajte LHS izraza in uporabite definicijo tangente:
LHS = (tanx + sinx) / (2tanx)
(sinx / cosx + sinx) / (2 (sinx / cosx)) t
(cosx / sinx) ((sinx / cosx + sinx) / 2) t
(cosx / sinx * sinx / cosx + cosx / sinx * sinx) / 2 t
(1 + cosx) / 2 t
Zdaj pa razmislite o RHS in uporabite identiteto:
cos2A - = 2cos ^ 2A - 1
Dajemo:
cosx - = 2cos ^ 2 (x / 2) - 1 => 1 + cosx - = 2cos ^ 2 (x / 2)
:. cos ^ 2 (x / 2) = (1 + cosx) / 2 = RHS
Tako:
# LHS = RHS => (tanx + sinx) / (2tanx) - = cos ^ 2 (x / 2) t QED
Kako dokazujete (cotx + cscx / sinx + tanx) = (cotx) (cscx)?
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Preverjeno spodaj (cotx + cscx) / (sinx + tanx) = (cotx) (cscx) (cosx / sinx + 1 / sinx) / (sinx + sinx / cosx) = (cotx) (cscx) ((cosx + 1) / ((sinxcosx) / cosx + sinx / cosx) = (cotx) (cscx) ((cosx + 1) / sinx) / ((sinx (cosx + 1)) / cosx) = (cotx) (cscx) ) (preklic (cosx + 1) / sinx) * (cosx / (sinxcancel ((cosx + 1)))))) (cotx) (cscx) (cosx / sinx * 1 / sinx) = (cotx) (cscx) ( cotx) (cscx) = (cotx) (cscx)
Kako dokazujete (cosx / (1 + sinx)) + ((1 + sinx) / cosx) = 2secx?
Pretvorite levo stran v izraze s skupnim imenovalcem in dodajte (pretvorite cos ^ 2 + sin ^ 2 v 1 na poti); poenostavimo in se sklicujemo na definicijo sek = 1 / cos (cos (x) / (1 + sin (x))) + ((1 + sin (x)) / cos (x)) = (cos ^ 2 (x) + 1 + 2sin (x) + sin ^ 2 (x)) / (cos (x) (1 + sin (x) = (2 + 2sin (x)) / (cos (x) (1 + sin (x)) ) = 2 / cos (x) = 2 * 1 / cos (x) = 2sek (x)
Kako dokazujete: secx - cosx = sinx tanx?
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S pomočjo definicij sekx in tanx, skupaj z identiteto sin ^ 2x + cos ^ 2x = 1, imamo secx-cosx = 1 / cosx-cosx = 1 / cosx-cos ^ 2x / cosx = (1-cos ^ 2x) / cosx = sin ^ 2x / cosx = sinx * sinx / cosx = sinxtanx