9. The goal of the following exercise is to prove that (sin x) = cos r. dr (a) By multiplying the numerator and denominator of the fraction on the left side by cos 0+ 1, then using an appropriate trigonometric identity, show that cos 0 – 1 sin 0 sin 0 (4) cos 0 +1* (b) Recall that lim,0 sine 1. Use the result in part (a) to show that %3D cos 0 – 1 lim (5) = 0 (c) Recall that sin(0, + 02) expand sin(r +h) in computing the derivative of f(x) = sin r. = sin 0, cos 02 + cos 01 sin 02. Use this identity to %3D (d) Show that the difference quotient f(x+h)-f(x) cos h - = sin x + cos x h (6) (e) Show that f'(x) = cos x by using equation (6) and limit properties. %3D
9. The goal of the following exercise is to prove that (sin x) = cos r. dr (a) By multiplying the numerator and denominator of the fraction on the left side by cos 0+ 1, then using an appropriate trigonometric identity, show that cos 0 – 1 sin 0 sin 0 (4) cos 0 +1* (b) Recall that lim,0 sine 1. Use the result in part (a) to show that %3D cos 0 – 1 lim (5) = 0 (c) Recall that sin(0, + 02) expand sin(r +h) in computing the derivative of f(x) = sin r. = sin 0, cos 02 + cos 01 sin 02. Use this identity to %3D (d) Show that the difference quotient f(x+h)-f(x) cos h - = sin x + cos x h (6) (e) Show that f'(x) = cos x by using equation (6) and limit properties. %3D
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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Transcribed Image Text:9. The goal of the following exercise is to prove that (sin r) = cos r.
(a) By multiplying the numerator and denominator of the fraction on the left
side by cos 0+1, then using an appropriate trigonometric identity, show that
cos 0 – 1
sin 0
sin 0
(4)
cos 0 +1*
(b) Recall that lim,0 sine = 1. Use the result in part (a) to show that
%3D
cos 8 – 1
lim
0-0
= 0
(5)
(c) Recall that sin(0, +02) = sin 0, cos 02 + cos 0, sin 02. Use this identity to
expand sin(x + h) in computing the derivative of f(x) = sin x.
(d) Show that the difference quotient
f(x+h)-f(x)
cos h -
sin h
= sin x
+ cos x
h
(6)
h
(e) Show that f'(x)
= cos x by using equation (6) and limit properties.
10. Do the following.
(a) Use the identity cos r sin (- x), the fact that (sin a)
chain rule to prove that (cos x) = - sin x.
= cos a, and the
(b) Use the quotient rule to prove that (tan x)
= sec? x. Use the identity
tan z = sin z
COS Z
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