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To demonstrate that the equation [tex]\((2 \cos \theta + 1)(2 \cos \theta - 1)(2 \cos 2 \theta - 1) = 2 \cos 4 \theta + 1\)[/tex] holds true, let's follow a detailed, step-by-step process to simplify and verify both sides of the equation.
### Step 1: Recognize the Trigonometric Identities
First, recall some useful trigonometric identities:
1. [tex]\( \cos(2\theta) = 2 \cos^2(\theta) - 1 \)[/tex]
2. [tex]\( \cos(4\theta) = 2 \cos^2(2\theta) - 1 \)[/tex]
3. We also know the double-angle identities can help:
- [tex]\( \cos(2\theta) = 2 \cos^2(\theta) - 1 \)[/tex]
### Step 2: Simplify the Left-Hand Side
Consider the left-hand side of the equation:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1)(2 \cos 2 \theta - 1) \][/tex]
First, simplify the product [tex]\( (2 \cos \theta + 1)(2 \cos \theta - 1) \)[/tex]:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = (2 \cos \theta)^2 - 1^2 = 4 \cos^2 \theta - 1 \][/tex]
Now, rewrite the left-hand side with this result:
[tex]\[ (4 \cos^2 \theta - 1)(2 \cos 2 \theta - 1) \][/tex]
Next, use the identity for [tex]\( \cos(2\theta) \)[/tex]:
[tex]\[ 2 \cos 2 \theta - 1 \][/tex]
Substitute [tex]\( \cos(2\theta) \)[/tex]:
[tex]\[ 2(2 \cos^2 \theta - 1) - 1 = 4 \cos^2 \theta - 2 - 1 = 4 \cos^2 \theta - 3 \][/tex]
Thus, our left-hand side now reads:
[tex]\[ (4 \cos^2 \theta - 1)(4 \cos^2 \theta - 3) \][/tex]
### Step 3: Simplify the Product
Notice that:
[tex]\[ (4 \cos^2 \theta - 1)(4 \cos^2 \theta - 3) = 16 \cos^4 \theta - 4 \cos^2 \theta - 12 \cos^2 \theta + 3 \][/tex]
[tex]\[ = 16 \cos^4 \theta - 16 \cos^2 \theta + 3 \][/tex]
### Step 4: Verify the Right-Hand Side
The right-hand side is:
[tex]\[ 2 \cos 4 \theta + 1 \][/tex]
Recall that:
[tex]\[ \cos(4\theta) = 2 \cos^2(2\theta) - 1 \][/tex]
[tex]\[ = 2(2 \cos^2(\theta) - 1)^2 - 1 \][/tex]
[tex]\[ = 2(4 \cos^4(\theta) - 4 \cos^2(\theta) + 1) - 1 \][/tex]
[tex]\[ = 8 \cos^4(\theta) - 8 \cos^2(\theta) + 2 - 1 \][/tex]
[tex]\[ = 8 \cos^4(\theta) - 8 \cos^2(\theta) + 1 \][/tex]
Thus:
[tex]\[ 2 \cos 4 \theta + 1 \][/tex]
[tex]\[ 2 [8 \cos^4(\theta) - 8 \cos^2(\theta) + 1] + 1\][/tex]
[tex]\[ = 16 \cos^4(\theta) - 16 \cos^2(\theta) + 2 + 1 \][/tex]
[tex]\[ = 16 \cos^4 \theta - 16 \cos^2 \theta + 3 \][/tex]
### Step 5: Conclusion
We've simplified both sides and see that:
[tex]\[ 16 \cos^4 \theta - 16 \cos^2 \theta + 3 = 16 \cos^4 \theta - 16 \cos^2 \theta + 3 \][/tex]
Therefore, both sides are equal, confirming the validity of the equation:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1)(2 \cos 2 \theta - 1) = 2 \cos 4 \theta + 1 \][/tex]
The equation holds true.
### Step 1: Recognize the Trigonometric Identities
First, recall some useful trigonometric identities:
1. [tex]\( \cos(2\theta) = 2 \cos^2(\theta) - 1 \)[/tex]
2. [tex]\( \cos(4\theta) = 2 \cos^2(2\theta) - 1 \)[/tex]
3. We also know the double-angle identities can help:
- [tex]\( \cos(2\theta) = 2 \cos^2(\theta) - 1 \)[/tex]
### Step 2: Simplify the Left-Hand Side
Consider the left-hand side of the equation:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1)(2 \cos 2 \theta - 1) \][/tex]
First, simplify the product [tex]\( (2 \cos \theta + 1)(2 \cos \theta - 1) \)[/tex]:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = (2 \cos \theta)^2 - 1^2 = 4 \cos^2 \theta - 1 \][/tex]
Now, rewrite the left-hand side with this result:
[tex]\[ (4 \cos^2 \theta - 1)(2 \cos 2 \theta - 1) \][/tex]
Next, use the identity for [tex]\( \cos(2\theta) \)[/tex]:
[tex]\[ 2 \cos 2 \theta - 1 \][/tex]
Substitute [tex]\( \cos(2\theta) \)[/tex]:
[tex]\[ 2(2 \cos^2 \theta - 1) - 1 = 4 \cos^2 \theta - 2 - 1 = 4 \cos^2 \theta - 3 \][/tex]
Thus, our left-hand side now reads:
[tex]\[ (4 \cos^2 \theta - 1)(4 \cos^2 \theta - 3) \][/tex]
### Step 3: Simplify the Product
Notice that:
[tex]\[ (4 \cos^2 \theta - 1)(4 \cos^2 \theta - 3) = 16 \cos^4 \theta - 4 \cos^2 \theta - 12 \cos^2 \theta + 3 \][/tex]
[tex]\[ = 16 \cos^4 \theta - 16 \cos^2 \theta + 3 \][/tex]
### Step 4: Verify the Right-Hand Side
The right-hand side is:
[tex]\[ 2 \cos 4 \theta + 1 \][/tex]
Recall that:
[tex]\[ \cos(4\theta) = 2 \cos^2(2\theta) - 1 \][/tex]
[tex]\[ = 2(2 \cos^2(\theta) - 1)^2 - 1 \][/tex]
[tex]\[ = 2(4 \cos^4(\theta) - 4 \cos^2(\theta) + 1) - 1 \][/tex]
[tex]\[ = 8 \cos^4(\theta) - 8 \cos^2(\theta) + 2 - 1 \][/tex]
[tex]\[ = 8 \cos^4(\theta) - 8 \cos^2(\theta) + 1 \][/tex]
Thus:
[tex]\[ 2 \cos 4 \theta + 1 \][/tex]
[tex]\[ 2 [8 \cos^4(\theta) - 8 \cos^2(\theta) + 1] + 1\][/tex]
[tex]\[ = 16 \cos^4(\theta) - 16 \cos^2(\theta) + 2 + 1 \][/tex]
[tex]\[ = 16 \cos^4 \theta - 16 \cos^2 \theta + 3 \][/tex]
### Step 5: Conclusion
We've simplified both sides and see that:
[tex]\[ 16 \cos^4 \theta - 16 \cos^2 \theta + 3 = 16 \cos^4 \theta - 16 \cos^2 \theta + 3 \][/tex]
Therefore, both sides are equal, confirming the validity of the equation:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1)(2 \cos 2 \theta - 1) = 2 \cos 4 \theta + 1 \][/tex]
The equation holds true.
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