Dokaži, da ((cos (33 ^ @)) ^ 2- (cos (57 ^ @)) ^ 2) / ((sin (10.5 ^ @)) ^ 2- (sin (34.5 ^ @)) ^ 2) = -sqrt2?

Odgovor:

Glej spodaj.

Pojasnilo:

Uporabljamo formule (A) - # cosA = sin (90 ^ @ - A) #, (B) - # cos ^ 2A-sin ^ 2A = cos2A #

(C) - # 2sinAcosA = sin2A #, (D) - # sinA + sinB = 2sin ((A + B) / 2) cos ((A-B) / 2) # in

(E) - # sinA-sinB = 2cos ((A + B) / 2) sin ((A-B) / 2) #

# (cos ^ 2 33 ^ @ - cos ^ 2 57 ^ @) / (sin ^ 2 10.5 ^ 2 ^ ^ ^ ^ 34,5 ^ @) #

= # (cos ^ 2 33 ^ @ - sin ^ 2 (90 ^ @ - 57 ^ @)) / ((sin10.5 ^ @ + sin34.5 ^ @) (sin10.5 ^ @ - sin34.5 ^ @)) # - uporabljeno A

= # (cos ^ 2 33 ^ @ - sin ^ 2 33 ^ @) / (- ((2sin22.5 ^ e ^ ^ ^ ^ @ @) (2cos22.5^@sin12 ^ @)) # - uporabljeno D & E

= # (cos66 ^ @) / (- (2sin22.5 ^ @ cos22.5 ^ @ xx2sin12 ^ @ cos12 ^ @) # - uporabljeno B

= # - (sin (90 ^ @ - 66 ^ @)) / (sin45 ^ @ sin24 ^ @) # - uporabljeno A & C

= # -sin24 ^ @ / (1 / sqrt2sin24 ^ @) #

= # -sqrt2 #