If `8tanx=15` , then `sinx-cosx` is equal to

Options

A. `8/17`
B. `17/7`
C. `1/17`
D. `7/17`

Answer

To solve the problem where \( 8 \tan x = 15 \) and we need to find \( \sin x - \cos x \), we can follow these steps:

### Step 1: Express \( \tan x \)
From the equation \( 8 \tan x = 15 \), we can express \( \tan x \) as:
\[ \tan x = \frac{15}{8} \]

### Step 2: Set Up the Right Triangle
In a right triangle, \( \tan x \) is defined as the ratio of the opposite side (perpendicular) to the adjacent side (base). Here, we can let:
- Opposite side (perpendicular) = 15
- Adjacent side (base) = 8

### Step 3: Calculate the Hypotenuse
Using the Pythagorean theorem, we can find the hypotenuse \( h \):
\[ h = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{15^2 + 8^2} \]
Calculating this:
\[ h = \sqrt{225 + 64} = \sqrt{289} = 17 \]

### Step 4: Find \( \sin x \) and \( \cos x \)
Now that we have the lengths of all sides of the triangle, we can find \( \sin x \) and \( \cos x \):
\[ \sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{15}{17} \]
\[ \cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{17} \]

### Step 5: Calculate \( \sin x - \cos x \)
Now we can find \( \sin x - \cos x \):
\[ \sin x - \cos x = \frac{15}{17} - \frac{8}{17} = \frac{15 - 8}{17} = \frac{7}{17} \]

### Final Answer
Thus, \( \sin x - \cos x = \frac{7}{17} \).

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