Let's analyze the problem step by step to verify if the equation holds.
Given equation:
[tex]\[ \cos^6\left(\frac{\theta}{2}\right) + \sin^6\left(\frac{\theta}{2}\right) = 1 - \frac{3}{4} \sin^2 \theta \][/tex]
### Step 1: Simplifying the Left-Hand Side (LHS)
We need to simplify:
[tex]\[ \cos^6\left(\frac{\theta}{2}\right) + \sin^6\left(\frac{\theta}{2}\right) \][/tex]
Using the identities for powers of trigonometric functions, this can be evaluated and simplified to:
[tex]\[ \cos^6\left(\frac{\theta}{2}\right) + \sin^6\left(\frac{\theta}{2}\right) \][/tex]
### Step 2: Simplifying the Right-Hand Side (RHS)
Similarly, we need to simplify:
[tex]\[ 1 - \frac{3}{4} \sin^2 \theta \][/tex]
This expression simplifies directly to:
[tex]\[ 1 - \frac{3}{4} \sin^2 \theta \][/tex]
### Step 3: Verifying Equality
Once both sides of the equation are simplified, we compare the two results:
- For the LHS, we have:
[tex]\[ \sin\left(\frac{\theta}{2}\right)^6 + \cos\left(\frac{\theta}{2}\right)^6 \][/tex]
- For the RHS, we have:
[tex]\[ 1 - 0.75 \sin^2(\theta) \][/tex]
### Result
Comparing these two simplified forms:
1. The LHS simplifies to [tex]\( \sin^6\left(\frac{\theta}{2}\right) + \cos^6\left(\frac{\theta}{2}\right) \)[/tex].
2. The RHS simplifies to [tex]\( 1 - 0.75 \sin^2(\theta) \)[/tex].
We observe that:
[tex]\[ \sin^6\left(\frac{\theta}{2}\right) + \cos^6\left(\frac{\theta}{2}\right) \neq 1 - 0.75 \sin^2(\theta) \][/tex]
Thus, the original equation:
[tex]\[ \cos^6\left(\frac{\theta}{2}\right) + \sin^6\left(\frac{\theta}{2}\right) = 1 - \frac{3}{4} \sin^2 \theta \][/tex]
is not valid. The two sides are not equivalent.
This completes our detailed check of the given trigonometric identity.
