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Cosec 48 + Cosec 96 + Cosec 192 + Cosec 384 = ? Please Explain

Cosec 48 + Cosec 96 + Cosec 192 + Cosec 384 = ?
Please Explain 
 

Grade:12th pass

2 Answers

Vikas TU
14149 Points
4 years ago
Dear Student 
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
Jitender Saini
23 Points
4 years ago
cosec 48 + cosec 96 = 1/cos 42 - 1/cos 78

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