Grade 12th passTrigonometry
Cosec 48 + Cosec 96 + Cosec 192 + Cosec 384 = ?Please Explain
Cosec 48 + Cosec 96 + Cosec 192 + Cosec 384 = ?
Please Explain
2 Answers
Vikas TU
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
cosec 48 + cosec 96 = 1/cos 42 - 1/cos 78
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