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Find the sum of the following series sinx​*cosec2x*cosec3x + sinx*cosec3x*cosec4x + sinx*cosec4x*cosec5x + ......to n terms

Find the sum of the following series
sinx​*cosec2x*cosec3x + sinx*cosec3x*cosec4x + sinx*cosec4x*cosec5x + ......to n terms

Grade:12th pass

1 Answers

Ritesh Khatri
76 Points
3 years ago
Tr = sinx*cosec(r+1)x*cosec(r+2)x
= sinx/[sin(r+1)x*sin(r+2)x]
={sin[(r+2)x-(r+1)x]}//[sin(r+1)x*sin(r+2)x]
=[sin(r+2)xcos(r+1)x – cos(r+2)xsin(r+1)x]/[sin(r+1)x*sin(r+2)x]
=cot(r+1)x – cot(r+2)x
Now substitute r= 1, 2 ,3 , ….. ,n and add all terms
we will get cot2x – cot(n+2)x

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