If `sec^(2)theta+tan^(2)theta=(7)/(12)`, then `sec^(4)theta-tan^(4)theta=?`

Options

A. `(7)/(12)`
B. `(1)/(2)`
C. `(5)/(12)`
D. 1

Answer

To solve the problem, we start with the given equation:

\[ \sec^2 \theta + \tan^2 \theta = \frac{7}{12} \]

We need to find the value of:

\[ \sec^4 \theta - \tan^4 \theta \]

### Step 1: Use the identity for the difference of squares
We can use the identity for the difference of squares, which states:

\[ a^2 - b^2 = (a + b)(a - b) \]

In our case, we can let \( a = \sec^2 \theta \) and \( b = \tan^2 \theta \). Therefore, we can rewrite:

\[ \sec^4 \theta - \tan^4 \theta = (\sec^2 \theta + \tan^2 \theta)(\sec^2 \theta - \tan^2 \theta) \]

### Step 2: Substitute the known value
From the problem, we know:

\[ \sec^2 \theta + \tan^2 \theta = \frac{7}{12} \]

So we can substitute this into our equation:

\[ \sec^4 \theta - \tan^4 \theta = \left(\frac{7}{12}\right)(\sec^2 \theta - \tan^2 \theta) \]

### Step 3: Find \(\sec^2 \theta - \tan^2 \theta\)
We can use the identity:

\[ \sec^2 \theta - \tan^2 \theta = 1 \]

This is derived from the Pythagorean identity:

\[ \sec^2 \theta = 1 + \tan^2 \theta \]

Thus,

\[ \sec^2 \theta - \tan^2 \theta = 1 \]

### Step 4: Substitute back into the equation
Now, substituting this back into our equation gives:

\[ \sec^4 \theta - \tan^4 \theta = \left(\frac{7}{12}\right)(1) = \frac{7}{12} \]

### Final Answer
Therefore, the value of \(\sec^4 \theta - \tan^4 \theta\) is:

\[ \frac{7}{12} \]

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