https://file.aiquickdraw.com/mathbot/file/17281351948215e5xrwm0.jpg

The equation in the image is:

\[
\cos^2 \theta + 3 \sin^2 \theta = 3 - 2 \cos^2 \theta
\]

Let's solve this step by step.

### Step 1: Simplify the equation

Start by moving all terms involving \(\cos^2 \theta\) and \(\sin^2 \theta\) to one side:

\[
\cos^2 \theta + 3 \sin^2 \theta + 2 \cos^2 \theta = 3
\]

Simplify the terms involving \(\cos^2 \theta\):

\[
3 \cos^2 \theta + 3 \sin^2 \theta = 3
\]

### Step 2: Use the Pythagorean identity

We know that:

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

So replace \(\sin^2 \theta + \cos^2 \theta\) with 1:

\[
3 (\sin^2 \theta + \cos^2 \theta) = 3
\]

Thus:

\[
3 \times 1 = 3
\]

This equation holds true, which means the original equation is an identity and is true for all values of \(\theta\).

### Conclusion:

The given equation is valid for all \(\theta\), meaning it is an identity.

Would you like a deeper explanation on any of the steps? Here are some related questions to explore:

1. What is the Pythagorean identity in trigonometry?
2. How can we solve trigonometric equations with multiple terms involving sine and cosine?
3. What are trigonometric identities and how are they derived?
4. Can identities like this be graphed? What would the graph of this equation look like?
5. How do you prove a trigonometric identity step-by-step?

**Tip:** When dealing with trigonometric identities, always try to recognize basic identities like the Pythagorean identity, as they can greatly simplify the equation.

Solve the trigonometric equation: cos^2(θ) + 3sin^2(θ) = 3 - 2cos^2(θ).

Learn how to solve the trigonometric equation cos²(θ) + 3sin²(θ) = 3 - 2cos²(θ) with a detailed step-by-step solution. This problem covers core trigonometric concepts, such as using the Pythagorean identity to simplify the equation. Suitable for students in Grades 10-12.

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