Answer :
Let's begin by analyzing the given equation and breaking it down step by step.
We need to prove:
[tex]\[ \cos^3(A) \cdot \cos(3A) + \sin^3(A) \cdot \sin(3A) = \cos^3(2A) \][/tex]
First, let's consider the left-hand side:
[tex]\[ \cos^3(A) \cdot \cos(3A) + \sin^3(A) \cdot \sin(3A) \][/tex]
### Simplifying the Left-Hand Side
Using trigonometric identities, we simplify the left-hand side further. Notice that from the trigonometric identities, we have:
- \(\cos(3A) = 4\cos^3(A) - 3\cos(A)\)
- \(\sin(3A) = 3\sin(A) - 4\sin^3(A)\)
Substituting these into our equation, we get:
[tex]\[ \cos^3(A) \cdot (4\cos^3(A) - 3\cos(A)) + \sin^3(A) \cdot (3\sin(A) - 4\sin^3(A)) \][/tex]
Expanding this, we obtain:
[tex]\[ \cos^3(A) \cdot 4\cos^3(A) - \cos^3(A) \cdot 3\cos(A) + \sin^3(A) \cdot 3\sin(A) - \sin^3(A) \cdot 4\sin^3(A) \][/tex]
Rewriting and combining like terms, we have:
[tex]\[ 4\cos^6(A) - 3\cos^4(A) + 3\sin^4(A) - 4\sin^6(A) \][/tex]
To simplify even further, consider the trigonometric identity:
- \(\cos(2A) = 2\cos^2(A) - 1\)
- \(\cos^3(2A) = \left(2\cos^2(A) - 1\right)^3\)
Notice that:
[tex]\[ 3\cos(2A) = \frac{3}{4}\cos(2A) + \frac{1}{4}\cos(6A) \][/tex]
This means we can rewrite the simplified form:
[tex]\[ 3\cos(2A)/4 + \cos(6A)/4 \][/tex]
### The Right-Hand Side
The right-hand side of the equation is:
[tex]\[ \cos^3(2A) \][/tex]
### Comparison
After simplification, if you compare the simplified left side and the right side, \(3\cos(2A)/4 + \cos(6A)/4 \neq \cos^3(2A)\). Hence, we can conclude that:
[tex]\[ \cos^3(A) \cdot \cos(3A) + \sin^3(A) \cdot \sin(3A) \neq \cos^3(2A) \][/tex]
Thus, the given equation is not an identity, as the simplified forms of the left and right sides are not equal.
We need to prove:
[tex]\[ \cos^3(A) \cdot \cos(3A) + \sin^3(A) \cdot \sin(3A) = \cos^3(2A) \][/tex]
First, let's consider the left-hand side:
[tex]\[ \cos^3(A) \cdot \cos(3A) + \sin^3(A) \cdot \sin(3A) \][/tex]
### Simplifying the Left-Hand Side
Using trigonometric identities, we simplify the left-hand side further. Notice that from the trigonometric identities, we have:
- \(\cos(3A) = 4\cos^3(A) - 3\cos(A)\)
- \(\sin(3A) = 3\sin(A) - 4\sin^3(A)\)
Substituting these into our equation, we get:
[tex]\[ \cos^3(A) \cdot (4\cos^3(A) - 3\cos(A)) + \sin^3(A) \cdot (3\sin(A) - 4\sin^3(A)) \][/tex]
Expanding this, we obtain:
[tex]\[ \cos^3(A) \cdot 4\cos^3(A) - \cos^3(A) \cdot 3\cos(A) + \sin^3(A) \cdot 3\sin(A) - \sin^3(A) \cdot 4\sin^3(A) \][/tex]
Rewriting and combining like terms, we have:
[tex]\[ 4\cos^6(A) - 3\cos^4(A) + 3\sin^4(A) - 4\sin^6(A) \][/tex]
To simplify even further, consider the trigonometric identity:
- \(\cos(2A) = 2\cos^2(A) - 1\)
- \(\cos^3(2A) = \left(2\cos^2(A) - 1\right)^3\)
Notice that:
[tex]\[ 3\cos(2A) = \frac{3}{4}\cos(2A) + \frac{1}{4}\cos(6A) \][/tex]
This means we can rewrite the simplified form:
[tex]\[ 3\cos(2A)/4 + \cos(6A)/4 \][/tex]
### The Right-Hand Side
The right-hand side of the equation is:
[tex]\[ \cos^3(2A) \][/tex]
### Comparison
After simplification, if you compare the simplified left side and the right side, \(3\cos(2A)/4 + \cos(6A)/4 \neq \cos^3(2A)\). Hence, we can conclude that:
[tex]\[ \cos^3(A) \cdot \cos(3A) + \sin^3(A) \cdot \sin(3A) \neq \cos^3(2A) \][/tex]
Thus, the given equation is not an identity, as the simplified forms of the left and right sides are not equal.
