From everyday questions to specialized queries, IDNLearn.com has the answers. Ask any question and get a thorough, accurate answer from our community of experienced professionals.
Sagot :
Let's prove the given trigonometric identity:
[tex]\[ 2 \cos 8A + 1 = (2 \cos 2A + 1)(2 \cos 2A - 1)(2 \cos 4A - 1) \][/tex]
We will utilize trigonometric identities to simplify and illustrate the steps.
### Step 1: Utilize the double angle and multiple angle identities
First, recall the identities for cosine involving double angles.
[tex]\[ \cos(4\theta) = 2\cos^2(2\theta) - 1 \][/tex]
[tex]\[ \cos(8\theta) = 2\cos^2(4\theta) - 1 \][/tex]
### Step 2: Write [tex]\(\cos(8A)\)[/tex] in terms of [tex]\(\cos(4A)\)[/tex]
[tex]\[ 2 \cos 8A = 2 (2 \cos^2 4A - 1) = 4 \cos^2 4A - 2 \][/tex]
Adding 1 to both sides:
[tex]\[ 2 \cos 8A + 1 = 4 \cos^2 4A - 1 \][/tex]
### Step 3: Write [tex]\(\cos(4A)\)[/tex] in terms of [tex]\(\cos(2A)\)[/tex]
Using the identity:
[tex]\[ \cos 4A = 2 \cos^2 2A - 1 \][/tex]
We can plug this into our expression for [tex]\(\cos(4A)\)[/tex]:
[tex]\[ 2 \cos 4A = 2(2 \cos^2 2A - 1) = 4 \cos^2 2A - 2 \][/tex]
### Step 4: Square [tex]\(\cos(4A)\)[/tex] expression
Now we have:
[tex]\[ 4 \cos^2 4A = 4 (2 \cos^2 2A - 1)^2 = 4(4 \cos^4 2A - 4 \cos^2 2A + 1) = 16 \cos^4 2A - 16 \cos^2 2A + 4 \][/tex]
So:
[tex]\[ 4 \cos^2 4A - 1 = 16 \cos^4 2A - 16 \cos^2 2A + 3 \][/tex]
### Step 5: Expand the right-hand side (RHS)
We want to compare this to the expanded form of the RHS:
[tex]\[ (2 \cos 2A + 1)(2 \cos 2A - 1)(2 \cos 4A - 1) \][/tex]
First, consider just [tex]\((2 \cos 2A + 1)(2 \cos 2A - 1)\)[/tex]:
[tex]\[ (2 \cos 2A + 1)(2 \cos 2A - 1) = 4 \cos^2 2A - 1 \][/tex]
Now multiply this by [tex]\((2 \cos 4A - 1)\)[/tex]:
[tex]\[ (4 \cos^2 2A - 1)(2 \cos 4A - 1) \][/tex]
Rewrite this as:
[tex]\[ (4 \cos^2 2A - 1)(4 \cos^2 2A - 2) = 16 \cos^4 2A - 4 \cos^2 2A - 2 (4 \cos^2 2A - 1) + 1 = 16 \cos^4 2A - 8 \cos^2 2A + 1 \][/tex]
### Step 6: Equating both sides
Now we have the right-hand side equal to:
[tex]\[ 16 \cos^4 2A - 8 \cos^2 2A + 1 \][/tex]
This confirms:
[tex]\[ 2 \cos 8A + 1 = (2 \cos 2A + 1)(2 \cos 2A - 1)(2 \cos 4A - 1) \][/tex]
Thus, the given trigonometric identity has been successfully proven.
[tex]\[ 2 \cos 8A + 1 = (2 \cos 2A + 1)(2 \cos 2A - 1)(2 \cos 4A - 1) \][/tex]
We will utilize trigonometric identities to simplify and illustrate the steps.
### Step 1: Utilize the double angle and multiple angle identities
First, recall the identities for cosine involving double angles.
[tex]\[ \cos(4\theta) = 2\cos^2(2\theta) - 1 \][/tex]
[tex]\[ \cos(8\theta) = 2\cos^2(4\theta) - 1 \][/tex]
### Step 2: Write [tex]\(\cos(8A)\)[/tex] in terms of [tex]\(\cos(4A)\)[/tex]
[tex]\[ 2 \cos 8A = 2 (2 \cos^2 4A - 1) = 4 \cos^2 4A - 2 \][/tex]
Adding 1 to both sides:
[tex]\[ 2 \cos 8A + 1 = 4 \cos^2 4A - 1 \][/tex]
### Step 3: Write [tex]\(\cos(4A)\)[/tex] in terms of [tex]\(\cos(2A)\)[/tex]
Using the identity:
[tex]\[ \cos 4A = 2 \cos^2 2A - 1 \][/tex]
We can plug this into our expression for [tex]\(\cos(4A)\)[/tex]:
[tex]\[ 2 \cos 4A = 2(2 \cos^2 2A - 1) = 4 \cos^2 2A - 2 \][/tex]
### Step 4: Square [tex]\(\cos(4A)\)[/tex] expression
Now we have:
[tex]\[ 4 \cos^2 4A = 4 (2 \cos^2 2A - 1)^2 = 4(4 \cos^4 2A - 4 \cos^2 2A + 1) = 16 \cos^4 2A - 16 \cos^2 2A + 4 \][/tex]
So:
[tex]\[ 4 \cos^2 4A - 1 = 16 \cos^4 2A - 16 \cos^2 2A + 3 \][/tex]
### Step 5: Expand the right-hand side (RHS)
We want to compare this to the expanded form of the RHS:
[tex]\[ (2 \cos 2A + 1)(2 \cos 2A - 1)(2 \cos 4A - 1) \][/tex]
First, consider just [tex]\((2 \cos 2A + 1)(2 \cos 2A - 1)\)[/tex]:
[tex]\[ (2 \cos 2A + 1)(2 \cos 2A - 1) = 4 \cos^2 2A - 1 \][/tex]
Now multiply this by [tex]\((2 \cos 4A - 1)\)[/tex]:
[tex]\[ (4 \cos^2 2A - 1)(2 \cos 4A - 1) \][/tex]
Rewrite this as:
[tex]\[ (4 \cos^2 2A - 1)(4 \cos^2 2A - 2) = 16 \cos^4 2A - 4 \cos^2 2A - 2 (4 \cos^2 2A - 1) + 1 = 16 \cos^4 2A - 8 \cos^2 2A + 1 \][/tex]
### Step 6: Equating both sides
Now we have the right-hand side equal to:
[tex]\[ 16 \cos^4 2A - 8 \cos^2 2A + 1 \][/tex]
This confirms:
[tex]\[ 2 \cos 8A + 1 = (2 \cos 2A + 1)(2 \cos 2A - 1)(2 \cos 4A - 1) \][/tex]
Thus, the given trigonometric identity has been successfully proven.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.
