To solve the equation \( \sin x - \cos x = 0 \), let's follow these steps:

### Step 1: Solve for \( \sin x \)

From the equation \( \sin x - \cos x = 0 \), we can rewrite it as:
\[
\sin x = \cos x
\]

### Step 2: Divide both sides by \( \cos x \) (assuming \( \cos x \neq 0 \))
\[
\frac{\sin x}{\cos x} = 1
\]

This simplifies to:
\[
\tan x = 1
\]

### Step 3: Solve for \( x \)

We know that \( \tan x = 1 \) at specific angles. The general solution for this is:
\[
x = \frac{\pi}{4} + n\pi \quad \text{where } n \text{ is any integer}.
\]

### Step 4: Calculate \( \sin x \) and \( \sec x \)

For the principal value \( x = \frac{\pi}{4} \):
- \( \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \)
- \( \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \)

Now, we can find the value of \( \sec x \), which is the reciprocal of \( \cos x \):
\[
\sec \left( \frac{\pi}{4} \right) = \frac{1}{\cos \left( \frac{\pi}{4} \right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}
\]

### Final Answers:
- \( \sin x = \frac{\sqrt{2}}{2} \)
- \( \sec x = \sqrt{2} \)

Would you like more details on any step?

### Related questions:
1. What is the value of \( \tan x \) if \( \sin x = \frac{\sqrt{2}}{2} \)?
2. How do you derive the general solution for \( \tan x = 1 \)?
3. What is the relation between \( \sin x \) and \( \sec x \)?
4. How can we use unit circle to find exact trigonometric values?
5. How does the identity \( \sin^2 x + \cos^2 x = 1 \) apply to this problem?

**Tip:** Always remember the Pythagorean identity, \( \sin^2 x + \cos^2 x = 1 \), to help solve trigonometric equations.

If sin x - cos x = 0, find sec x and sin x.

Learn how to solve the trigonometric equation sin x - cos x = 0 and find sec x. This step-by-step guide covers key trigonometric concepts, including the use of identities and the unit circle. Ideal for students in Grades 10-12.

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