Proving the Trigonometric Identity: (sin^4 x cos^4 x 1 - 2sin^2 x cos^2 x)

In this guide, we will walk through the steps to prove the trigonometric identity (sin^4 x cos^4 x 1 - 2sin^2 x cos^2 x). This identity is important in trigonometry and is often used in various mathematical proofs and problem solving.

Step 1: Understanding the Left-Hand Side

The left-hand side of the identity is (sin^4 x cos^4 x). We can rewrite this expression using algebraic identities to simplify the proof process.

Option 1: Using Algebraic Identities

Let's use the identity (a^2b^2 a^2b - 2ab). Here, let (a sin^2 x) and (b cos^2 x). Then:

(sin^4 x cos^4 x (sin^2 x)^2 (cos^2 x)^2 (sin^2 x)(cos^2 x)^2 - 2(sin^2 x)(cos^2 x))

Simplifying further, we get:

(sin^4 x cos^4 x sin^2 x cos^4 x - 2sin^2 x cos^2 x)

Step 2: Simplifying the Right-Hand Side

The right-hand side of the identity is (1 - 2sin^2 x cos^2 x). To show that both sides are equal, we will manipulate the right-hand side to match the left-hand side.

First, we can start by rewriting the right-hand side: Note that (sin^2 x csc^2 x 1), so we can write:

(sin^2 x csc^2 x - 2cos^2 x)

Step 3: Proving Both Sides Are Equal

Next, we will show that the left-hand side is equivalent to the right-hand side by simplifying both expressions.

Option 2: Using Cosecant and Trigonometric Identities

Let's start by rewriting the right-hand side for a clear comparison:

(1 - 2 sin^2 x cos^2 x 1 - 2 sin^2 x cos^2 x)

Now, let's simplify the left-hand side:

(sin^4 x cos^4 x 2 sin^2 x cos^2 x - 2 sin^2 x cos^2 x (sin^2 x cos^2 x)^2 - 2 sin^2 x cos^2 x)

Simplifying further using the known identity (sin^2 x cos^2 x 1):

(1 - 2 sin^2 x cos^2 x sin^2 x csc^2 x - 2 cos^2 x)

This simplifies to:

(1 - 2 sin^2 x cos^2 x)

Which is exactly the same as the right-hand side of the original equation.

Conclusion

We have shown that:

(sin^4 x cos^4 x 1 - 2sin^2 x cos^2 x)

This identity is a useful tool in trigonometry and helps in simplifying complex expressions. By using algebraic manipulation and trigonometric identities, we can prove that both sides of the equation are indeed equal.

Keywords

Trigonometric identity, proving trigonometric identities, algebraic manipulation