What is the value of `((tan^2 x - sin^2 x))/(sec^2x)` ?

Options

A. `sin^4 x`
B. `cos^2 x`
C. `sin^2 x`
D. `cos^4 x`

Answer

To solve the expression \(\frac{\tan^2 x - \sin^2 x}{\sec^2 x}\), we will follow these steps:

### Step 1: Rewrite \(\tan^2 x\) and \(\sec^2 x\) in terms of \(\sin x\) and \(\cos x\)
We know that:
\[ \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \]
and
\[ \sec^2 x = \frac{1}{\cos^2 x} \]

### Step 2: Substitute these identities into the expression
Substituting these identities into the expression gives us:
\[ \frac{\frac{\sin^2 x}{\cos^2 x} - \sin^2 x}{\frac{1}{\cos^2 x}} \]

### Step 3: Simplify the numerator
To simplify the numerator:
\[ \frac{\sin^2 x}{\cos^2 x} - \sin^2 x = \frac{\sin^2 x - \sin^2 x \cos^2 x}{\cos^2 x} \]
Now factor out \(\sin^2 x\):
\[ \frac{\sin^2 x (1 - \cos^2 x)}{\cos^2 x} \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity \(1 - \cos^2 x = \sin^2 x\), we can substitute:
\[ \frac{\sin^2 x \cdot \sin^2 x}{\cos^2 x} = \frac{\sin^4 x}{\cos^2 x} \]

### Step 5: Divide by \(\sec^2 x\)
Now, we can rewrite the entire expression:
\[ \frac{\sin^4 x}{\cos^2 x} \cdot \cos^2 x = \sin^4 x \]

### Final Answer
Thus, the value of the expression \(\frac{\tan^2 x - \sin^2 x}{\sec^2 x}\) is:
\[ \sin^4 x \]