Answer :

Certainly! Let's verify the trigonometric identity [tex]\(\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)\)[/tex] step-by-step.

### Step 1: Express [tex]\(\sin(3\theta)\)[/tex] using a known trigonometric identity

The triple angle formula for the sine function is a well-known trigonometric identity:
[tex]\[ \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \][/tex]

### Step 2: Verify the identity

We'll use this triple angle formula directly:
[tex]\[ \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \][/tex]

The expression on the right-hand side is exactly what we have:
[tex]\[ 3\sin(\theta) - 4\sin^3(\theta) \][/tex]

So, according to the given trigonometric identity, this should be identical to [tex]\(\sin(3\theta)\)[/tex].

### Step 3: Simplify and equate

By directly applying the trigonometric identity:
[tex]\[ \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \][/tex]

Given that both sides are representations of the same trigonometric function, we conclude that the identity holds true.

Thus, the expression [tex]\(\sin(3\theta)\)[/tex] is indeed [tex]\(3\sin(\theta) - 4\sin^3(\theta)\)[/tex], and this verifies the trigonometric identity.

### Final Verification

As per our verification, it is clear that:
[tex]\[ \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \][/tex]

The identity holds true, and you can confidently use this result in your trigonometric calculations.