Prove that e²1+²2 e²1e²2 for all 21, 22 € C. Hint: The function ƒ(z) = e²eª-≈ has ƒ' (z) = 0 for any fixed complex number a € C. Use arguments from functions of one variable to show that f is constant (=eª) on C. Show that Show that = cos²z + sin² z = 1, for all z e C. sin(2₁ +22) = sin(z₁) cos(z2) + sin(z2) cos(z₁), for all 2₁, 22 € C.

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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(a) Prove that e²¹+²² = e²¹ ²² for all 21,22 € C.
Hint: The function ƒ(z) = e²eª-² has ƒ'(z) = 0 for any fixed complex number
a € C. Use arguments from functions of one variable to show that f is constant
(= eª) on C.
(b) Show that
(c) Show that
cos²z + sin²z =
COS
1, for all
ZEC.
sin(21 +22) = sin(21) cos(22) + sin(22) cos(21), for all
21, 22 € C.
Transcribed Image Text:(a) Prove that e²¹+²² = e²¹ ²² for all 21,22 € C. Hint: The function ƒ(z) = e²eª-² has ƒ'(z) = 0 for any fixed complex number a € C. Use arguments from functions of one variable to show that f is constant (= eª) on C. (b) Show that (c) Show that cos²z + sin²z = COS 1, for all ZEC. sin(21 +22) = sin(21) cos(22) + sin(22) cos(21), for all 21, 22 € C.
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