`(sec^(2)theta-1)/(tan^(2)theta)=?`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the expression \((\sec^2 \theta - 1) / \tan^2 \theta\), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be rearranged to express \(\sec^2 \theta\):
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

### Step 2: Substitute \(\sec^2 \theta\) in the Expression
Now, substitute \(\sec^2 \theta\) in the original expression:
\[ \frac{\sec^2 \theta - 1}{\tan^2 \theta} = \frac{(1 + \tan^2 \theta) - 1}{\tan^2 \theta} \]

### Step 3: Simplify the Numerator
The numerator simplifies as follows:
\[ (1 + \tan^2 \theta) - 1 = \tan^2 \theta \]
So, we can rewrite the expression:
\[ \frac{\tan^2 \theta}{\tan^2 \theta} \]

### Step 4: Simplify the Fraction
Now, simplifying the fraction gives us:
\[ \frac{\tan^2 \theta}{\tan^2 \theta} = 1 \]

### Final Answer
Thus, the value of \((\sec^2 \theta - 1) / \tan^2 \theta\) is:
\[ \boxed{1} \]