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The given equation is:

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

We are tasked to solve for \(\sec^2\theta + \cos^2\theta - 1\).

### Step 1: Simplify the given equation
\[
2\sin^2\theta - \cos^2\theta + 2 = 2
\]
Subtract 2 from both sides:
\[
2\sin^2\theta - \cos^2\theta = 0
\]

### Step 2: Use the identity \(\sin^2\theta + \cos^2\theta = 1\)
We know that:
\[
\sin^2\theta = 1 - \cos^2\theta
\]
Substitute this into the equation:
\[
2(1 - \cos^2\theta) - \cos^2\theta = 0
\]
Expanding the terms:
\[
2 - 2\cos^2\theta - \cos^2\theta = 0
\]
Simplify:
\[
2 - 3\cos^2\theta = 0
\]
Solving for \(\cos^2\theta\):
\[
3\cos^2\theta = 2 \quad \Rightarrow \quad \cos^2\theta = \frac{2}{3}
\]

### Step 3: Find \(\sec^2\theta\)
We know that:
\[
\sec^2\theta = \frac{1}{\cos^2\theta}
\]
Substitute \(\cos^2\theta = \frac{2}{3}\):
\[
\sec^2\theta = \frac{1}{\frac{2}{3}} = \frac{3}{2}
\]

### Step 4: Calculate \(\sec^2\theta + \cos^2\theta - 1\)
Now, substitute the values of \(\sec^2\theta = \frac{3}{2}\) and \(\cos^2\theta = \frac{2}{3}\) into the expression:
\[
\sec^2\theta + \cos^2\theta - 1 = \frac{3}{2} + \frac{2}{3} - 1
\]

To simplify, find a common denominator:
\[
\sec^2\theta + \cos^2\theta - 1 = \frac{9}{6} + \frac{4}{6} - \frac{6}{6} = \frac{9 + 4 - 6}{6} = \frac{7}{6}
\]

Thus, the value of \(\sec^2\theta + \cos^2\theta - 1\) is \(\frac{7}{6}\).

Would you like more details or have any questions?

Here are some related questions to explore:
1. What is the identity involving \(\sec^2\theta\) and \(\tan^2\theta\)?
2. How can you express \(\sec^2\theta\) in terms of \(\sin^2\theta\)?
3. How do other trigonometric identities help simplify equations like this?
4. What are the Pythagorean identities in trigonometry?
5. How do you solve trigonometric equations using factoring?

**Tip:** Always check if trigonometric identities like \(\sin^2\theta + \cos^2\theta = 1\) can simplify complex expressions.

2 sin^2(θ) - (cos^2(θ) - 2) = 2. Find sec^2(θ) + cos^2(θ) - 1

Solve the trigonometric equation 2 sin^2(θ) - (cos^2(θ) - 2) = 2 by simplifying using key identities like sin^2(θ) + cos^2(θ) = 1. Learn to calculate sec^2(θ) + cos^2(θ) - 1 with step-by-step solutions for students in Grades 10-12.

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