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Solve using the 8 basic trig identities and 1 side

The given trigonometric equation is:

\[
\frac{2 \sin^4 A + \sin^2 A \cos^2 A - \cos^4 A}{3 \sin^2 A - 1} = 1
\]

Let's simplify this step by step, using trigonometric identities where possible. We'll rely on the Pythagorean identity and other basic identities.

### Step 1: Apply Pythagorean identities

We know the following identities:
1. \(\sin^2 A + \cos^2 A = 1\)
2. \(\cos^2 A = 1 - \sin^2 A\) (from the first identity)

Let's substitute \(\cos^2 A = 1 - \sin^2 A\) into the equation wherever \(\cos^2 A\) or \(\cos^4 A\) appears.

### Step 2: Express \(\cos^4 A\)

We can write \(\cos^4 A\) as:
\[
\cos^4 A = (\cos^2 A)^2 = (1 - \sin^2 A)^2
\]
Now expand \((1 - \sin^2 A)^2\):
\[
\cos^4 A = 1 - 2\sin^2 A + \sin^4 A
\]

### Step 3: Substitute into the numerator

Now, substitute this into the numerator:
\[
2 \sin^4 A + \sin^2 A \cos^2 A - \cos^4 A = 2 \sin^4 A + \sin^2 A (1 - \sin^2 A) - (1 - 2\sin^2 A + \sin^4 A)
\]
Simplify this expression:
\[
= 2 \sin^4 A + \sin^2 A - \sin^4 A - (1 - 2\sin^2 A + \sin^4 A)
\]
\[
= 2 \sin^4 A + \sin^2 A - \sin^4 A - 1 + 2 \sin^2 A - \sin^4 A
\]
\[
= 2 \sin^4 A - \sin^4 A - \sin^4 A + \sin^2 A + 2 \sin^2 A - 1
\]
\[
= 0 \sin^4 A + 3 \sin^2 A - 1
\]
Thus, the numerator simplifies to:
\[
3 \sin^2 A - 1
\]

### Step 4: Simplify the denominator

The denominator is already \(3 \sin^2 A - 1\).

### Step 5: Simplify the equation

Now, the equation becomes:
\[
\frac{3 \sin^2 A - 1}{3 \sin^2 A - 1} = 1
\]
This simplifies directly to:
\[
1 = 1
\]

Thus, the given equation holds true. The equation is an identity.

Would you like more details on this solution or have any further questions?

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Here are 5 questions that can help you explore related topics:
1. What are the 8 basic trigonometric identities, and how can they be applied in other contexts?
2. How can the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\) be used to simplify complex trigonometric expressions?
3. How do you expand and simplify expressions like \((1 - \sin^2 A)^2\)?
4. What is the significance of verifying trigonometric identities in equations?
5. How can you solve trigonometric equations when the identity is not immediately obvious?

**Tip**: When simplifying trigonometric identities, always look for opportunities to use basic identities like the Pythagorean identities early in the process. They often lead to simpler expressions quickly.

Learn how to solve a complex trigonometric equation using basic identities like the Pythagorean identity. This solution includes step-by-step simplifications and is suitable for students in Grades 10-12.

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