If `theta` be acute angle and `costheta=(15)/(17)` , then the value of `cot(90^(@)-theta)` is

Options

A. `(2sqrt(8))/(15)`
B. `(8)/(15)`
C. `(sqrt(2))/(17)`
D. `(8sqrt(2))/(17)`

Answer

To find the value of \( \cot(90^\circ - \theta) \) given that \( \cos \theta = \frac{15}{17} \), we can follow these steps:

### Step 1: Understand the relationship
We know that:
\[ \cot(90^\circ - \theta) = \tan \theta \]
This means we need to find \( \tan \theta \).

### Step 2: Use the cosine value to find the sides of the triangle
Given \( \cos \theta = \frac{15}{17} \), we can interpret this in terms of a right triangle where:
- The adjacent side (base) is 15
- The hypotenuse is 17

### Step 3: Find the opposite side using the Pythagorean theorem
Using the Pythagorean theorem:
\[ h^2 = p^2 + b^2 \]
where \( h \) is the hypotenuse, \( p \) is the opposite side, and \( b \) is the adjacent side. Plugging in our values:
\[ 17^2 = p^2 + 15^2 \]
Calculating the squares:
\[ 289 = p^2 + 225 \]
Now, isolate \( p^2 \):
\[ p^2 = 289 - 225 = 64 \]
Taking the square root gives:
\[ p = \sqrt{64} = 8 \]

### Step 4: Calculate \( \tan \theta \)
Now we can find \( \tan \theta \):
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{15} \]

### Step 5: Find \( \cot(90^\circ - \theta) \)
Since \( \cot(90^\circ - \theta) = \tan \theta \), we have:
\[ \cot(90^\circ - \theta) = \frac{8}{15} \]

### Final Answer
Thus, the value of \( \cot(90^\circ - \theta) \) is:
\[ \frac{8}{15} \]

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