Kako dokazujete, da sqrt (3) cos (x + pi / 6) - cos (x + pi / 3) = cos (x) -sqrt3sinx?

# LHS = sqrt3cos (x + pi / 6) -cos (x-pi / 3) #

# = sqrt3 cosx * cos (pi / 6) -sinx * sin (pi / 6) - cosx * cos (pi / 3) -sinx * sin (pi / 3) #

# = sqrt3 cosx * (sqrt3 / 2) -sinx * (1/2) - cosx * (1/2) -sinx * (sqrt3 / 2) #

# = (3cosx-sqrt3sinx) / 2- (cosx-sqrt3sinx) / 2 #

# = (3cosx-sqrt3sinx-cosx + sqrt3sinx) / 2 #

# = (2cosx) / 2 = cosx = RHS #

FCF (Funkcionalna nadaljevalna frakcija) cosh_ (cf) (x; a) = cosh (x + a / cosh (x + a / cosh (x + ...))). Kako dokazujete, da je ta FCF parna funkcija glede na x in a, skupaj? In cosh_ (cf) (x; a) in cosh_ (cf) (-x; a) sta različni?

Cosh_ (cf) (x; a) = cosh_ (cf) (- x; a) in cosh_ (cf) (x; -a) = cosh_ (cf) (- x; -a). Ker so cosh vrednosti> = 1, je vsako y tukaj> = 1 Pokažimo, da je y = cosh (x + 1 / y) = cosh (-x + 1 / y). Ustrezne dve strukturi FCF sta različni. Graf za y = cosh (x + 1 / y). Opazujte, da je a = 1, x> = - 1 graf {x-ln (y + (y ^ 2-1) ^ 0.5) + 1 / y = 0} Graf za y = cosh (-x + 1 / y). Opazujte, da je a = 1, x <= 1 graf {x + ln (y + (y ^ 2-1) ^ 0.5) -1 / y = 0} kombinirani graf za y = cosh (x + 1 / y) in y = cosh (-x + 1 / y): graf {(x-ln (y + (y ^ 2-1) ^ 0.5) + 1 / y) (x + ln (y + (y ^ 2-1) ^ 0.5) - 1 / y) = 0}. Prav tako je

Kaj je (sqrt (5+) sqrt (3)) / (sqrt (3+) sqrt (3+) sqrt (5)) - (sqrt (5-) sqrt (3)) / (sqrt (3+) sqrt (3-) sqrt (5))?

2/7 vzamemo, A = (sqrt5 + sqrt3) / (sqrt3 + sqrt3 + sqrt5) - (sqrt5-sqrt3) / (sqrt3 + sqrt3-sqrt5) = (sqrt5 + sqrt3) / (2sqrt3 + sqrt5) - (sqrt5) -sqrt3) / (2sqrt3-sqrt5) = (sqrt5-sqrt3) / (2sqrt3-sqrt5) = ((sqrt5 + sqrt3) (2sqrt3-sqrt5) - (sqrt5-sqrt3) ) (2sqrt3 + sqrt5) ((2sqrt15-5 + 2 * 3-sqrt15) - (2sqrt15-5 + 2 * 3-sqrt15)) / ((2sqrt3)) ^ 2- (sqrt5) ^ 2) = (prekliči (2sqrt15) -5 + 2 * 3zaključi (-sqrt15) - prekliči (2sqrt15) -5 + 2 * 3 + prekliči (sqrt15)) / (12-5) = ( -10 + 12) / 7 = 2/7 Upoštevajte, da če je v imenovalcu (sqrt3 + sqrt (3 + sqrt5)) in (sqrt3 + sqrt (3-sqrt5)), bo odgovor spremenjen.

Kako poenostavite (1 / sqrt (a-1) + sqrt (a + 1)) / (1 / sqrt (a + 1) -1 / sqrt (a-1)) div sqrt (a + 1) / ( (a-1) sqrt (a + 1) - (a + 1) sqrt (a-1)), a> 1?

Ogromno oblikovanje matematike ...> barva (modra) (((1 / sqrt (a-1) + sqrt (a + 1)) / (1 / sqrt (a + 1) -1 / sqrt (a-1)) ) / (sqrt (a + 1) / ((a-1) sqrt (a + 1) - (a + 1) sqrt (a-1))) = barva (rdeča) (((1 / sqrt (a- 1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) / (sqrt (a + 1) cdot sqrt (a-1)))) / (sqrt (a +1) / (sqrt (a-1) cdot sqrt (a-1) cdot sqrt (a + 1) -sqrt (a + 1) cdot sqrt (a + 1) sqrt (a-1))) = barva ( modro) ((((1 / sqrt (a-1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) / (sqrt (a + 1) cdot sqrt (a -1)))) / (sqrt (a + 1) / (sqrt (a + 1) cdot sqrt (a-1) (sqrt (a-1) -sqrt (a + 1))) = barva (rdeča) ((1 / sqr