8 tan 𝑥 = 15, then sin𝑥 − cos𝑥 is equal to 
(a) 7
17
 (b) 8
17
 (c) 7
4
 (d) 64
49

We are given the equation:

\[
8 \tan x = 15
\]

To find \( \sin x - \cos x \), let's proceed step by step.

### Step 1: Express \( \tan x \) in terms of sine and cosine
We know that:
\[
\tan x = \frac{\sin x}{\cos x}
\]

Using this in the given equation, we have:
\[
8 \frac{\sin x}{\cos x} = 15
\]

Simplifying this:
\[
\frac{\sin x}{\cos x} = \frac{15}{8}
\]

### Step 2: Use the identity for \( \sin^2 x + \cos^2 x = 1 \)
Let \( \sin x = 15k \) and \( \cos x = 8k \), where \( k \) is a constant. This ensures that the ratio \( \frac{\sin x}{\cos x} = \frac{15}{8} \).

Now, apply the Pythagorean identity:
\[
\sin^2 x + \cos^2 x = 1
\]

Substitute \( \sin x = 15k \) and \( \cos x = 8k \):
\[
(15k)^2 + (8k)^2 = 1
\]
\[
225k^2 + 64k^2 = 1
\]
\[
289k^2 = 1
\]
\[
k^2 = \frac{1}{289}
\]
\[
k = \frac{1}{17}
\]

### Step 3: Calculate \( \sin x - \cos x \)
Now that we have \( k = \frac{1}{17} \), we can find \( \sin x \) and \( \cos x \):
\[
\sin x = 15k = \frac{15}{17}, \quad \cos x = 8k = \frac{8}{17}
\]

Thus:
\[
\sin x - \cos x = \frac{15}{17} - \frac{8}{17} = \frac{7}{17}
\]

### Final Answer:
\[
\boxed{\frac{7}{17}} \quad \text{(Option a)}
\]

Would you like a deeper explanation on any part of this process?

Here are 5 related questions to explore:
1. How do you derive the tangent from sine and cosine?
2. Why does the identity \( \sin^2 x + \cos^2 x = 1 \) hold true?
3. What are the other trigonometric identities involving sine and cosine?
4. How can you use tangent values to find sine and cosine in general?
5. Can you explain the Pythagorean identity in more detail?

**Tip:** Always check for unit circle values when dealing with standard trigonometric problems for quicker solutions.

8 tan 𝑥 = 15, then sin𝑥 − cos𝑥 is equal to (a) 7/17 (b) 8/17 (c) 7/4 (d) 64/49

Step-by-step solution to the problem 8 tan x = 15, where we calculate sin x - cos x using trigonometric identities and ratios. Learn about the tangent, sine, cosine, and the Pythagorean Identity. Suitable for Grades 10-12 students looking to understand key trigonometry concepts.

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