Apakah fungsi sebenar ialah (e ^ (ix) -e ^ (- ix)) / (ie ^ (ix) + ie ^ (- ix)) sama dengan?

Jawapan:

#tan (x) #

Penjelasan:

# e ^ (ix) = cos (x) + i sin (x) #

#cos (-x) = cos (x) #

#sin (-x) = -sin (x) #

Jadi:

(ix) -e ^ (- ix) = (cos (x) + i sin (x)) - (cos (-x) + i sin (-x)

# = (cos (x) + i sin (x)) - (cos (x) -i sin (x)) = 2i sin (x) #

Dan:

(ix) + e ^ (- ix) = (cos (x) + i sin (x)) + (cos (-x) + i sin (-x)

# = (cos (x) + i sin (x)) + (cos (x) -i sin (x)) = 2 cos (x) #

Jadi:

(x) = (2i cos (x)) = dosa (x) / cos (x) = tan (x) #