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EXAMPLE 5 Prove cos 4x = 8 cos“ x – 8 cos? x + 1. PROOF We can write cos 4x as cos (2 · 2x) and apply our double-angle for- mula. Because the right side is written in terms of cos x only, we will choose the second form of our double-angle formula for cos 2A. = cos (2 · 2x) = 2 cos 2x – 1 = 2(2 cos? x – 1)² – 1 = 2(4 cos* x – 4 cos? x + 1) – 1 = 8 cos x - 8 cos x + 2 – 1 = 8 cos“ x – 8 cos? x + 1 cos 4x Double-angle formula Double-angle formula %3D Square || Distribute Simplify |

EXAMPLE 5 Prove cos 4x = 8 cos“ x – 8 cos? x + 1. PROOF We can write cos 4x as cos (2 · 2x) and apply our double-angle for- mula. Because the right side is written in terms of cos x only, we will choose the second form of our double-angle formula for cos 2A. = cos (2 · 2x) = 2 cos 2x – 1 = 2(2 cos? x – 1)² – 1 = 2(4 cos* x – 4 cos? x + 1) – 1 = 8 cos x - 8 cos x + 2 – 1 = 8 cos“ x – 8 cos? x + 1 cos 4x Double-angle formula Double-angle formula %3D Square || Distribute Simplify |

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Hello, I don't understand this part of chapter 5.3 (attached picture) 

I get how we get cos4x = cos (2*2x)

But cost of 2x = 2*(cos^2)x-1, not 2*(cos^2)2x-1

So I had figured this would be 2(2cos^2 x-1)

but then I don't get how we got to the part the arrow points to where we're squaring that whole thing?  Could someone please explain step by step?

 

Thank you!

EXAMPLE 5 Prove cos 4x = 8 cos“ x – 8 cos? x + 1. - PROOF We can write cos 4x as cos (2· 2x) and apply our double-angle for- mula. Because the right side is written in terms of cos x only, we will choose the second form of our double-angle formula for cos 2A. cos 4x = cos (2 · 2x) = 2 cos? 2x - 1 2(2 cos² x – 1)² – 1 = 2(4 cos“ x – 4 cos? x + 1) – 1 = 8 cos* x – 8 cos x + 2 - 1 = 8 cos“ x – 8 cos² x + 1 Double-angle formula Double-angle formula - X Distribute 8 cos x + 1 Simplify
Transcribed Image Text:EXAMPLE 5 Prove cos 4x = 8 cos“ x – 8 cos? x + 1. - PROOF We can write cos 4x as cos (2· 2x) and apply our double-angle for- mula. Because the right side is written in terms of cos x only, we will choose the second form of our double-angle formula for cos 2A. cos 4x = cos (2 · 2x) = 2 cos? 2x - 1 2(2 cos² x – 1)² – 1 = 2(4 cos“ x – 4 cos? x + 1) – 1 = 8 cos* x – 8 cos x + 2 - 1 = 8 cos“ x – 8 cos² x + 1 Double-angle formula Double-angle formula - X Distribute 8 cos x + 1 Simplify
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