To solve the equation:

\[
\sin(3x - 25^\circ) - \cos(x + 15^\circ) = 0
\]

### Step 1: Express both trigonometric functions in terms of sine.
We know the identity:

\[
\cos(\theta) = \sin\left(90^\circ - \theta\right)
\]

Applying this identity to \(\cos(x + 15^\circ)\), we get:

\[
\cos(x + 15^\circ) = \sin\left(90^\circ - (x + 15^\circ)\right) = \sin(75^\circ - x)
\]

Thus, the equation becomes:

\[
\sin(3x - 25^\circ) - \sin(75^\circ - x) = 0
\]

### Step 2: Use the sine subtraction identity.
We use the identity for \(\sin A - \sin B\):

\[
\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)
\]

Let \(A = 3x - 25^\circ\) and \(B = 75^\circ - x\), then:

\[
\sin(3x - 25^\circ) - \sin(75^\circ - x) = 2 \cos\left(\frac{(3x - 25^\circ) + (75^\circ - x)}{2}\right) \sin\left(\frac{(3x - 25^\circ) - (75^\circ - x)}{2}\right)
\]

Simplifying:

\[
= 2 \cos\left(\frac{3x - 25^\circ + 75^\circ - x}{2}\right) \sin\left(\frac{3x - 25^\circ - 75^\circ + x}{2}\right)
\]

This simplifies further to:

\[
= 2 \cos\left(\frac{2x + 50^\circ}{2}\right) \sin\left(\frac{2x - 100^\circ}{2}\right)
\]

\[
= 2 \cos(x + 25^\circ) \sin(x - 50^\circ)
\]

### Step 3: Set the equation to zero.
Now we have:

\[
2 \cos(x + 25^\circ) \sin(x - 50^\circ) = 0
\]

For this equation to be true, either:

1. \(\cos(x + 25^\circ) = 0\), or
2. \(\sin(x - 50^\circ) = 0\)

### Step 4: Solve each case.

#### Case 1: \(\cos(x + 25^\circ) = 0\)
The cosine function is zero at odd multiples of \(90^\circ\):

\[
x + 25^\circ = 90^\circ + 180^\circ n, \quad n \in \mathbb{Z}
\]

Thus:

\[
x = 65^\circ + 180^\circ n
\]

#### Case 2: \(\sin(x - 50^\circ) = 0\)
The sine function is zero at multiples of \(180^\circ\):

\[
x - 50^\circ = 180^\circ n, \quad n \in \mathbb{Z}
\]

Thus:

\[
x = 50^\circ + 180^\circ n
\]

### Final solutions:

The solutions for \(x\) are:

\[
x = 65^\circ + 180^\circ n \quad \text{or} \quad x = 50^\circ + 180^\circ n, \quad n \in \mathbb{Z}
\]

Let me know if you need further details or explanations.

### Related questions:
1. How do we prove the identity \(\cos(\theta) = \sin(90^\circ - \theta)\)?
2. How do we apply the sine subtraction identity \(\sin A - \sin B\)?
3. What are the general solutions for \(\cos(\theta) = 0\)?
4. How do we determine general solutions for \(\sin(\theta) = 0\)?
5. What are the properties of trigonometric functions with respect to shifts and translations?

### Tip:
Always check for extraneous solutions when solving trigonometric equations, as certain steps might introduce invalid solutions.

Learn how to solve the trigonometric equation Sin(3x - 25°) - Cos(x + 15°) = 0 step-by-step using key trigonometric identities. This problem explores the cosine-sine identity and the sine subtraction formula. Suitable for students in Grades 11-12.

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