Bevis at Cos ^ 6 (x) + sin ^ 6 (x) = 1/8 (5 + 3cos4x)?

Vi bruger

# Rarra ^ 3 + b ^ 3 = (a + b) (a ^ 2-ab + b ^ 2) #

# Rarra ^ 2 + b ^ 2 = (a-b) ^ 2 + 2ab #

# Rarrsin ^ 2x + cos ^ 2x = 1 #

# rarr2cos ^ 2x = 1 + cos2x # og

# Rarr2sin ^ 2x = 1-cos2x #

# LHS = cos ^ 6 (x) + sin ^ 6 (x) #

# = (Cos ^ 2x) ^ 3 + (sin ^ 2x) ^ 3 #

# = Cos ^ 2x + sin ^ 2 x (cos ^ 2x) ^ 2-cos ^ 2x * sin ^ 2x + sin ^ 2x) ^ 2 #

# = 1 * (cos ^ 2x-sin ^ 2x) ^ 2 + 2cos ^ 2x * sin ^ 2x-cos ^ 2x * sin ^ 2 x #

# = Cos ^ 2 (2x) + cos ^ 2x * sin ^ 2 x #

# = 1/4 4cos ^ 2 (2x) + 4cos ^ 2x * sin ^ 2 x #

# = 1/4 2 (1 + cos4x) + sin ^ 2 (2x) #

# = 2 / (4 * 2) 2 + 2cos4x + sin ^ 2 (2x) #

# = 1/8 4 + 4cos4x + 2sin ^ 2 (2x) #

# = 1/8 4 + 4cos4x + 1-cos4x #

# = 1/8 5 + 3cos4x = RHS #

Bevis at ((cos (33 ^ @)) 2- (cos (57 ^ @)) 2) / ((sin (10,5 ^ @)) 2- (sin (34,5 ^ @)) 2) = -sqrt2?

Se nedenfor. Vi bruger formler (A) - cosA = sin (90 ^ @ - A), (B) - cos ^ 2A-sin ^ 2A = cos2A (C) -2sinAcosA = sin2A, (D) - sinA + sinB = 2sin A + B) / 2) cos ((AB) / 2) og (E) - sinA-sinB = 2cos ((A + B) / 2) sin ((AB) / 2) cos ^ 2 57 ^ @) / (sin ^ 2 10.52@-sin2 34.5 ^ @) = (cos ^ 2 33 ^ @ sin ^ 2 (90 ^ @ 57 ^ @)) ((sin10. 5 ^ @ + sin34.5 ^ @) (sin10.5 ^ @ sin34.5 ^ @)) - anvendt A = (cos ^ 2 33 ^ @ sin ^ 2 33 ^ @) / (- (2sin22.5 ^^ cos12 ^ @) (2cos22.5 ^ @ sin12 ^ @)) - anvendt D & E = (cos66 ^ @) / (- (2sin22.5 ^ @ cos22.5 ^ xx2sin12 ^ @ cos12 ^ @) - anvendt B = - (sin (90 ^ @ - 66 ^ @)) (sin45 ^ @ sin24 ^ @) - anve

Bevis dette: (1-sin ^ 4x-cos ^ 4x) / (1-sin ^ 6x-cos ^ 6x) = 2/3

LHS = (1-sin ^ 4x-cos ^ 4x) / (1-sin ^ 6x-cos ^ 6x) = (1 - ((sin ^ 2x) ^ 2 + (cos ^ 2x) ^ 2)) / - ((sin ^ 2x) ^ 3 + (cos ^ 2x) ^ 3)) = (1 - ((sin ^ 2x + cos ^ 2x) ^ 2-2sin ^ 2cos ^ 2x)) / (1 - ((sin ^ 2x + cos ^ 2x) ^ 3-3sin ^ 2xcos ^ 2x (sin ^ 2x + cos ^ 2x)) = (1- (sin ^ 2x + cos ^ 2x) ^ 2 + 2sin ^ 2cos ^ 2x) / (1 - (sin ^ 2x + cos ^ 2x) ^ 3 + 3sin ^ 2xcos ^ 2x (sin ^ 2x + cos ^ 2x)) = (1-1 ^ 2 + 2sin ^ 2cos ^ 2x) / (1-1 ^ 3 + 3sin ^ 2xcos ^ 2x) = (2sin ^ 2cos ^ 2x) / (3sin ^ 2xcos ^ 2x) = 2/3 = RHS Proved I trin 3 anvendes de følgende formler a ^ 2 + b ^ 2 = (a + b) ^ 2-2ab og a ^ 3 + b ^ 3 = (a + b) ^ 3-3ab (a + b

Bevis at Cot 4x (sin 5 x + sin 3 x) = Cot x (sin 5 x - sin 3 x)?

# sin a + sin b = 2 sin ((a + b) / 2) cos ((ab) / 2) sin a - sin b = 2 sin ((ab) / 2) cos ((a + b) / 2 ) Højre side: cot x (sin 5x - sin 3x) = cot x cdot 2 sin ((5x-3x) / 2) cos ((5x + 3x) / 2) = cos x / sin x cdot 2 sin x cos 4x = 2 cos x cos 4x venstre side: barneseng (4x) (sin 5x + sin 3x) = barneseng (4x) cdot 2 sin ((5x + 3x) / 2) cos ((5x-3x) / 2) = {cos 4x} / {sin 4x} cdot 2 sin 4x cos x = 2 cos x cos 4 x De er lige quad sqrt #