If Sin `x=4/5,` then `Sec^2x-1=?`

Options

A. `16/25`
B. `25/9`
C. `9/16`
D. `16/9`

Answer

To solve the problem where we need to find the value of \( \sec^2 x - 1 \) given that \( \sin x = \frac{4}{5} \), we can follow these steps:

### Step 1: Understand the relationship between sine and secant
We know that:
\[ \sec x = \frac{1}{\cos x} \]
and from the Pythagorean identity, we have:
\[ \sin^2 x + \cos^2 x = 1 \]

### Step 2: Calculate \( \cos x \)
Given \( \sin x = \frac{4}{5} \), we can find \( \cos x \) using the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting \( \sin x \):
\[ \left(\frac{4}{5}\right)^2 + \cos^2 x = 1 \]
\[ \frac{16}{25} + \cos^2 x = 1 \]
\[ \cos^2 x = 1 - \frac{16}{25} \]
\[ \cos^2 x = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]

### Step 3: Calculate \( \sec^2 x \)
Now that we have \( \cos^2 x \), we can find \( \sec^2 x \):
\[ \sec^2 x = \frac{1}{\cos^2 x} = \frac{1}{\frac{9}{25}} = \frac{25}{9} \]

### Step 4: Calculate \( \sec^2 x - 1 \)
Now we can find \( \sec^2 x - 1 \):
\[ \sec^2 x - 1 = \frac{25}{9} - 1 = \frac{25}{9} - \frac{9}{9} = \frac{25 - 9}{9} = \frac{16}{9} \]

### Final Answer
Thus, the value of \( \sec^2 x - 1 \) is:
\[ \frac{16}{9} \]

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