Kako dokazati Tan ^ 2 (x / 2 + Pi / 4) = (1 + sinx) / (1-sinx)?

Odgovor:

Dokazilo spodaj (dolgo je)

Pojasnilo:

To delam nazaj (toda pisanje vnaprej bi dobro delovalo):

# (1 + sinx) / (1-sinx) = (1 + sinx) / (1-sinx) * (1 + sinx) / (1 + sinx) #

# = (1 + sinx) ^ 2 / (1-sin ^ 2x) #

# = (1 + sinx) ^ 2 / cos ^ 2x #

# = ((1 + sinx) / cosx) ^ 2 #

Potem nadomestite # t # formula (Razlaga spodaj)

# = ((1+ (2t) / (1 + t ^ 2)) / ((1-t ^ 2) / (1 + t ^ 2))) ^ 2 #

# = (((1 + t ^ 2 + 2t) / (1 + t ^ 2)) / ((1-t ^ 2) / (1 + t ^ 2))) ^ 2 #

# = ((1 + t ^ 2 + 2t) / (1-t ^ 2)) ^ 2 #

# = ((1 + 2t + t ^ 2) / (1-t ^ 2)) ^ 2 #

# = ((1 + t) ^ 2 / (1-t ^ 2)) ^ 2 #

# = ((1 + t) ^ 2 / ((1-t) (1 + t))) ^ 2 #

# = ((1 + t) / (1-t)) ^ 2 #

# = ((1 + tan (x / 2)) / (1-tan (x / 2))) ^ 2 #

# = ((tan (pi / 4) + tan (x / 2)) / (1-tan (x / 2) tan (pi / 4))) ^ 2 # Upoštevajte, da: (#tan (pi / 4) = 1) #

# = (tan (x / 2 + pi / 4)) ^ 2 #

# = tan ^ 2 (x / 2 + pi / 4) #

T FORMULASE ZA TE ENAKOST:

# sinx = (2t) / (1-t ^ 2) #, # cosx = (1-t ^ 2) / (1 + t ^ 2) #, kje # t = tan (x / 2) #

Kako dokazati (1 + sinx-cosx) / (1 + cosx + sinx) = tan (x / 2)?

Glej spodaj. LHS = (1-cosx + sinx) / (1 + cosx + sinx) = (2sin ^ 2 (x / 2) + 2sin (x / 2) * cos (x / 2)) / (2cos ^ 2 (x / 2) + 2sin (x / 2) * cos (x / 2) = (2sin (x / 2) [sin (x / 2) + cos (x / 2)]) / (2cos (x / 2) * [ sin (x / 2) + cos (x / 2)]) = tan (x / 2) = RHS

Kako dokazati to identiteto? sin ^ 2x + tan ^ 2x * sin ^ 2x = tan ^ 2x

Spodaj prikazano ... Uporabite naše trigonomske lastnosti ... sin ^ 2 x + cos ^ 2 x = 1 => sin ^ 2 x / cos ^ 2 x + cos ^ 2 x / cos ^ 2 x = 1 / cos ^ 2 x => tan ^ 2 x + 1 = 1 / cos ^ 2 x faktor leva stran vaše težave ... => sin ^ 2 x (1 + tan ^ 2 x) => sin ^ 2 x (1 / cos) ^ 2 x) = sin ^ 2 x / cos ^ 2 x => (sinx / cosx) ^ 2 = tan ^ 2 x

Kako dokazati tan (x / 2) = sinx + cosxcotx-cotx?

Razvijte pravo stran. Vemo, da tan (x / 2) = (1 - cos (x)) / sin (x). Tako razvijamo pravo stran enakosti. cot (x) = 1 / tan (x) tako: sin (x) + cos (x) posteljica (x) - otroška posteljica (x) = (sin ^ 2 (x) + cos ^ 2 (x) - cos (x )) / sin (x) = (1-cos (x)) / sin (x) = tan (x / 2).