Sin^8x - cos^8x = (sin^2x - cos^2x)×(1-2sin^2x cos^2x)

To solve the equation \( \sin^8 x - \cos^8 x = (\sin^2 x - \cos^2 x)(1 - 2\sin^2 x \cos^2 x) \), we can start by applying some algebraic identities and transformations. This equation revolves around the difference of squares and the properties of trigonometric functions. Let's break it down step by step.

Understanding the Left Side

The left-hand side of the equation, \( \sin^8 x - \cos^8 x \), can be recognized as a difference of squares. We can factor this expression:

• Recall that \( a^2 - b^2 = (a - b)(a + b) \).
• In our case, let \( a = \sin^4 x \) and \( b = \cos^4 x \). Thus, we can rewrite it as:
• \( \sin^8 x - \cos^8 x = (\sin^4 x - \cos^4 x)(\sin^4 x + \cos^4 x) \).

Factoring Further

Next, we can further factor \( \sin^4 x - \cos^4 x \) using the same difference of squares technique:

• Set \( a = \sin^2 x \) and \( b = \cos^2 x \). Then, we have:
• \( \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) \).
• Since \( \sin^2 x + \cos^2 x = 1 \), this simplifies to:
• \( \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x) \cdot 1 = \sin^2 x - \cos^2 x \).

Combining It All

Now we can substitute back into the equation:

• We have \( \sin^8 x - \cos^8 x = (\sin^2 x - \cos^2 x)(\sin^4 x + \cos^4 x) \).
• Next, we need to find \( \sin^4 x + \cos^4 x \). This can be computed using the identity:
• \( \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \).

Finalizing the Equation

Now we can plug this back into our equation:

• Thus, \( \sin^8 x - \cos^8 x = (\sin^2 x - \cos^2 x)(1 - 2\sin^2 x \cos^2 x) \).

Conclusion

We have successfully shown that the left side \( \sin^8 x - \cos^8 x \) simplifies to the right side \( (\sin^2 x - \cos^2 x)(1 - 2\sin^2 x \cos^2 x) \). Both sides of the equation are equal, confirming that the initial equation holds true. This approach illustrates the power of algebraic identities in simplifying trigonometric expressions.