Bonus 1. (a) For z + 1, verify the identity1- zn+11- z1+ z + z²+... + z" =(b) Use part (a) and appropriate identities from complex numbers to establishLagrange's trigonometric identity11+ cos 0 + cos 20 + ... + cos no2sin (n +5)02 sin 0for 0 < 0 < 27.(c) For n an even positive integer, establish the formulancos non-2 0 sin² 0+4cos"-40 sin“ 0+.+(-1)"/2 sin" 0cos" 0–cosis the binomial coefficient defined bykwhereп(п — 1)(п — 2)... (п — k + 1)()kk!

(a) It is required to prove the identity For z≠1, 1+z+z2+ . . . +zn=1-zn+11-z…

1. (a) For z # 1, verify the identity 1– zn+1 1+z + z² +... + z" = 1- z (b) Use part (a) and appropriate identities from complex numbers to establish Lagrange's trigonometric identity 1 sin (n + )0 1+ cos 0 + cos 20 +... + cos n0 = + 2 sin 0 for 0 < 0 < 27. (c) For n an even positive integer, establish the formula cos no = cos" 0– cos"-2 0 sin? 0+ 2 cos"-4 0 sin“ 0+...+(-1)"/2 sin" 0 4 where k is the binomial coefficient defined by п(n - 1)(п — 2)..(п — k+1) k | k!

1. (a) For z # 1, verify the identity 1– zn+1 1+z + z² +... + z" = 1- z (b) Use part (a) and appropriate identities from complex numbers to establish Lagrange's trigonometric identity 1 sin (n + )0 1+ cos 0 + cos 20 +... + cos n0 = + 2 sin 0 for 0 < 0 < 27. (c) For n an even positive integer, establish the formula cos no = cos" 0– cos"-2 0 sin? 0+ 2 cos"-4 0 sin“ 0+...+(-1)"/2 sin" 0 4 where k is the binomial coefficient defined by п(n - 1)(п — 2)..(п — k+1) k | k!

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

100%

Bonus 1. (a) For z + 1, verify the identity
1- zn+1
1- z
1+ z + z²
+... + z" =
(b) Use part (a) and appropriate identities from complex numbers to establish
Lagrange's trigonometric identity
1
1+ cos 0 + cos 20 + ... + cos no
2
sin (n +5)0
2 sin 0
for 0 < 0 < 27.
(c) For n an even positive integer, establish the formula
n
cos no
n-2 0 sin² 0+
4
cos"-40 sin“ 0+.+(-1)"/2 sin" 0
cos" 0–
cos
is the binomial coefficient defined by
k
where
п(п — 1)(п — 2)... (п — k + 1)
()
k
k!

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