Given, cosec 48 + cosec 96 + cosec 192 + cosec 384

Now, cosec 48 = cosec (90 - 42) = sec 42 = 1/cos 42

and cosec 192 = cosec (270 - 78) = -sec 78 = -1/cos 78

So, cosec 48 + cosec 96 = 1/cos 42 - 1/cos 78

                                  = (cos 78 - cos 42)/(cos 78 * cos 42)

                                  = -(2* sin 60 * sin 18)/{cos (60 + 18) * cos (60 - 18)}     {Apply cos C - cos D formula}

                                  = -(2* sin 60 * sin 18)/{cos² 60 - sin² 18} 

                                  =  -(2* √3/2 * (√5 - 1)/4}/{(1/2)²  - ((√5 - 1)/4)² }

                                  = -2√3

Again,

Now, cosec 384 = cosec (360 + 24) = cosec 24

So, cosec 96 + cosec 24 = 1/sin 96 + 1/sin 24

                                  = (sin 96 + sin 24)/(sin 96 * sin 24)

                                  = (2* sin 60 * cos 36)/{sin (60 + 36) * sin (60 - 36)}     {Apply cos C - cos D formula}

                                  = (2* sin 60 * cos 36)/{sin² 60 - sin² 36} 

                                  =  (2* √3/2 * (√5 + 1)/4}/{(√3/2)²  - ((√5 + 1)/4)² }

                                  = 2√3

So, cosec 48 + cosec 96 + cosec 192 + cosec 384 = -2√3 + 2√3 = 0