If `cos^(4)theta-sin^(4)theta=(2)/(13)`, find `cos^(2)theta-sin^(2)theta+1`.

Answer

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

\[ \cos^4 \theta - \sin^4 \theta = \frac{2}{13} \]

### Step 1: Use the difference of squares identity

We can use the identity for the difference of squares, which states that \( a^2 - b^2 = (a - b)(a + b) \). Here, we can rewrite \(\cos^4 \theta - \sin^4 \theta\) as:

\[ (\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta) \]

### Step 2: Simplify using the Pythagorean identity

We know from the Pythagorean identity that:

\[ \cos^2 \theta + \sin^2 \theta = 1 \]

Substituting this into our equation gives:

\[ (\cos^2 \theta - \sin^2 \theta)(1) = \frac{2}{13} \]

Thus, we have:

\[ \cos^2 \theta - \sin^2 \theta = \frac{2}{13} \]

### Step 3: Find \(\cos^2 \theta - \sin^2 \theta + 1\)

Now, we need to find \(\cos^2 \theta - \sin^2 \theta + 1\):

\[ \cos^2 \theta - \sin^2 \theta + 1 = \frac{2}{13} + 1 \]

### Step 4: Convert 1 to a fraction

To add these fractions, we convert 1 into a fraction with a denominator of 13:

\[ 1 = \frac{13}{13} \]

### Step 5: Add the fractions

Now we can add the two fractions:

\[ \frac{2}{13} + \frac{13}{13} = \frac{2 + 13}{13} = \frac{15}{13} \]

### Final Answer

Thus, the final result is:

\[ \cos^2 \theta - \sin^2 \theta + 1 = \frac{15}{13} \]

---