To prove or disprove the identity [tex]\(\cos(4\theta) - 4\cos(2\theta) \equiv 8\sin^4(\theta) - 3\)[/tex], let's start by examining each side of the equation separately and see if they can be simplified or transformed to match each other.
Step 1: Simplify the right-hand side (RHS)
The RHS is [tex]\(8 \sin^4(\theta) - 3\)[/tex]. We can express [tex]\(\sin^4(\theta)\)[/tex] in terms of cosine.
First, recall the double angle identity:
[tex]\[ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \][/tex]
Squaring both sides to get [tex]\(\sin^4(\theta)\)[/tex]:
[tex]\[ \sin^4(\theta) = \left( \frac{1 - \cos(2\theta)}{2} \right)^2 \][/tex]
[tex]\[ = \frac{(1 - \cos(2\theta))^2}{4} \][/tex]
[tex]\[ = \frac{1 - 2\cos(2\theta) + \cos^2(2\theta)}{4} \][/tex]
Thus, the term [tex]\(8 \sin^4(\theta)\)[/tex] can be rewritten as:
[tex]\[ 8 \sin^4(\theta) = 8 \cdot \frac{1 - 2\cos(2\theta) + \cos^2(2\theta)}{4} \][/tex]
[tex]\[ = 2(1 - 2\cos(2\theta) + \cos^2(2\theta)) \][/tex]
[tex]\[ = 2 - 4\cos(2\theta) + 2\cos^2(2\theta) \][/tex]
Now include the [tex]\(-3\)[/tex] term:
[tex]\[ 8 \sin^4(\theta) - 3 = 2 - 4\cos(2\theta) + 2\cos^2(2\theta) - 3 \][/tex]
[tex]\[ = -1 - 4\cos(2\theta) + 2\cos^2(2\theta) \][/tex]
Step 2: Simplify the left-hand side (LHS)
The LHS is [tex]\(\cos(4\theta) - 4 \cos(2\theta)\)[/tex]. We know the double angle identity for cosine:
[tex]\[ \cos(4\theta) = 2 \cos^2(2\theta) - 1 \][/tex]
Using this in the LHS we get:
[tex]\[ \cos(4\theta) - 4 \cos(2\theta) = (2 \cos^2(2\theta) - 1) - 4 \cos(2\theta) \][/tex]
[tex]\[ = 2 \cos^2(2\theta) - 1 - 4 \cos(2\theta) \][/tex]
Step 3: Compare the simplified forms
From the simplifications:
LHS: [tex]\(2 \cos^2(2\theta) - 1 - 4 \cos(2\theta)\)[/tex]
RHS: [tex]\(-1 - 4\cos(2\theta) + 2\cos^2(2\theta)\)[/tex]
When we compare the LHS and RHS:
[tex]\[ 2 \cos^2(2\theta) - 1 - 4 \cos(2\theta) \equiv -1 - 4\cos(2\theta) + 2\cos^2(2\theta) \][/tex]
They are identical.
Thus, contrary to original assumption from verification, YES, the identity [tex]\(\cos(4\theta) - 4\cos(2\theta) \equiv 8\sin^4(\theta) - 3\)[/tex] holds true. So, [tex]\(\cos(4\theta) - 4 \cos(2\theta)\)[/tex] and [tex]\(8 \sin^4(\theta) - 3\)[/tex] are indeed equivalent.
