Hvordan beviser du at 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 (Functional Continued Fraction) cosh_ (cf) (x; a) = cosh (x + a / cosh (x + a / cosh (x + ...))). Hvordan beviser du at denne FCF er en jævn funktion med hensyn til både x og a, sammen? Og cosh_ (cf) (x; a) og cosh_ (cf) (-x; a) er forskellige?

Cosh_ (cf) (x; a) = cosh_ (cf) (- x; a) og cosh_ (cf) (x; -a) = cosh_ (cf) (- x; -a). Som cosh værdier er> = 1, nogen y her> = 1 Lad os vise, at y = cosh (x + 1 / y) = cosh (-x + 1 / y) Graferne er lavet tildele a = + -1. De tilsvarende to strukturer af FCF er forskellige. Graf for y = cosh (x + 1 / y). Bemærk, at a = 1, x> = - 1 graf {x-ln (y + (y ^ 2-1) ^ 0,5) + 1 / y = 0} Graf for y = cosh (-x + 1 / y). Bemærk, at a = 1, x <= 1 graf (x + ln (y + (y ^ 2-1) ^ 0,5) -1 / y = 0} Kombineret graf for y = cosh (x + 1 / y) og y = cosh (-x + 1 / y): graf {(x-ln (y + (y ^ 2-1) ^ 0,5) + 1 / y) (x + ln (y

Hvad er (sqrt (5+) sqrt (3)) / (sqrt (3+) sqrt (3+) sqrt (5)) - (sqrt (5-) sqrt (3)) / (3-) sqrt (5))?

2/7 Vi tager A = (sqrt5 + sqrt3) / (sqrt3 + sqrt3 + sqrt5) - (sqrt5-sqrt3) / (sqrt3 + sqrt3-sqrt5) = (sqrt5 + sqrt3) / (2sqrt3 + sqrt5) - -sqrt3) / (2sqrt3-sqrt5) = (sqrt5 + sqrt3) / (2sqrt3 + sqrt5) - (sqrt5-sqrt3) / (2sqrt3-sqrt5) = ((sqrt5 + sqrt3) (sqrt5-sqrt5) ) (2sqrt3 + sqrt5)) / ((2sqrt3 + sqrt5) (2sqrt3-sqrt5) = ((2sqrt15-5 + 2 * 3-sqrt15) - (2sqrt15 + 5-2 * 3-sqrt15)) / ((2sqrt3) ^ 2- (sqrt5) ^ 2) = (annullere (2sqrt15) -5 + 2 * 3cancel (-sqrt15) - annullere (2sqrt15) -5 + 2 * 3 + annullere (sqrt15)) / (12-5) = ( -10 + 12) / 7 = 2/7 Bemærk, at hvis i betegnelserne er (sqrt3 + sqrt (3 + sqrt5)) og (sqrt3 + sq

Hvordan forenkler du (1 / sqrt (a-1) + sqrt (a + 1)) / (1 / sqrt (a + 1) -1 / sqrt (a-1)) div sqrt (a-1) sqrt (a + 1) - (a + 1) sqrt (a-1)), a> 1?

Kæmpe matematisk formatering ...> farve (blå) ((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))) = farve (rød) 1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) / (sqrt (a + 1) cdot sqrt (a-1))) / +1) / (sqrt (a-1) cdot sqrt (a-1) cdot sqrt (a + 1) -sqrt (a + 1) cdot sqrt (a + 1) sqrt (a-1))) = farve blå) ((1 / sqrt (a-1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) / (sqrt (a + 1) cdot sqrt -1)))) (sqrt (a + 1) / (sqrt (a + 1) cdot sqrt (a-1) (sqrt (a-1) -sqrt (a + 1))) = farve / (Sqrt (a-1) -sqrt (a + 1)) / (sqrt (a +