If `sin x = (4)/(5) `, then cosec x + cot x =

Options

A. `31//12`
B. `35//12`
C. 2
D. `1//2`

Answer

To solve the problem where \( \sin x = \frac{4}{5} \) and we need to find \( \csc x + \cot x \), we can follow these steps:

### Step 1: Understand the relationship of sine, cosecant, and cotangent
We know that:
- \( \csc x = \frac{1}{\sin x} \)
- \( \cot x = \frac{\cos x}{\sin x} \)

### Step 2: Calculate cosecant
Given \( \sin x = \frac{4}{5} \):
\[ \csc x = \frac{1}{\sin x} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \]

### Step 3: Find the value of cosine using the Pythagorean theorem
Since \( \sin x = \frac{4}{5} \), we can find \( \cos x \) using the identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting the value of \( \sin x \):
\[ \left(\frac{4}{5}\right)^2 + \cos^2 x = 1 \]
\[ \frac{16}{25} + \cos^2 x = 1 \]
\[ \cos^2 x = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \cos x = \frac{3}{5} \]

### Step 4: Calculate cotangent
Now we can find \( \cot x \):
\[ \cot x = \frac{\cos x}{\sin x} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

### Step 5: Add cosecant and cotangent
Now we can find \( \csc x + \cot x \):
\[ \csc x + \cot x = \frac{5}{4} + \frac{3}{4} = \frac{5 + 3}{4} = \frac{8}{4} = 2 \]

### Final Answer
Thus, \( \csc x + \cot x = 2 \).

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