# Solve Sin(^6)x + Cos(^6)x Factoring Problem

• riddlingminion
In summary, to factor \sin^{6}x +\cos^{6}x, we can use the identity x^3+y^3=(x+y)(x^2-xy-y^2) to simplify the expression to (\sin^{2}x +\cos^{2}x)(\sin^{4}x +\cos^{4}x)-2\sin^{2}x \cos^{2}x(\sin^{2}x +\cos^{2}x). This can then be further simplified to \cos^{2} 2x. Additionally, the unknown equality \frac{1}{8}+\frac{1}{8}=0 is proven.
riddlingminion
How do I factor sin(^6)x + cos(^6)x?

Use x^3+y^3=(x+y)(x^2-xy-y^2).

Better

$$\sin^{6}x +\cos^{6}x= (\sin^{2}x +\cos^{2}x)(\sin^{4}x +\cos^{4}x)-2\sin^{2}x \cos^{2}x(\sin^{2}x +\cos^{2}x)=...=\cos^{2} 2x.$$

Daniel.

Doesn't $$( \sin^{2} x +\cos^{2} x ) ( \sin^{4}x +\cos^{4} x ) - 2 \sin^{2}x \cos^{2} x ( \sin^{2} x +\cos^{2} x) = \sin^6 x+ \cos^6 x - \sin^2 x \cos^4 x- \sin^4 x \cos^2 x$$?

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dextercioby said:
Better

$$\sin^{6}x +\cos^{6}x= (\sin^{2}x +\cos^{2}x)(\sin^{4}x +\cos^{4}x)-2\sin^{2}x \cos^{2}x(\sin^{2}x +\cos^{2}x)=...=\cos^{2} 2x.$$

Daniel.

Whereby the hitherto unknown equality:
$$\frac{1}{8}+\frac{1}{8}=0$$
is proven.

Yes, thought it was 2 simple 2 be true. You can drop that 2, then. :d

Daniel.

## 1. What is the formula for solving the trigonometric expression Sin(^6)x + Cos(^6)x?

The formula for solving Sin(^6)x + Cos(^6)x is Sin(^6)x + Cos(^6)x = (Sin(^2)x + Cos(^2)x)(Sin(^4)x - Sin(^2)x*Cos(^2)x + Cos(^4)x). This can be derived using the trigonometric identity (Sin(^2)x + Cos(^2)x) = 1 and the difference of squares formula.

## 2. How do I factor the expression Sin(^6)x + Cos(^6)x?

To factor Sin(^6)x + Cos(^6)x, you can use the formula mentioned above and rewrite it as (Sin(^2)x + Cos(^2)x)(Sin(^4)x - Sin(^2)x*Cos(^2)x + Cos(^4)x). Then, you can factor out the common term (Sin(^2)x + Cos(^2)x) to get (Sin(^2)x + Cos(^2)x)(Sin(^4)x - Sin(^2)x*Cos(^2)x + Cos(^4)x) = (Sin(^2)x + Cos(^2)x)(Sin(^2)x - Cos(^2)x)(Sin(^2)x + Cos(^2)x)(Sin(^2)x - Cos(^2)x). Finally, you can use the difference of squares formula again to factor the remaining terms and simplify the expression.

## 3. Can I use a calculator to solve the expression Sin(^6)x + Cos(^6)x?

Yes, you can use a calculator to solve the expression Sin(^6)x + Cos(^6)x. However, it is recommended to understand the formula and factor it manually to improve your understanding of trigonometric identities and expressions.

## 4. Are there any special cases when solving Sin(^6)x + Cos(^6)x?

Yes, there is a special case when solving Sin(^6)x + Cos(^6)x. If x = π/2 + kπ (where k is any integer), then the expression becomes Sin(^6)(π/2 + kπ) + Cos(^6)(π/2 + kπ) = (-1)^6 + (0)^6 = 1. In this case, the expression cannot be factored further.

## 5. Can I use the formula for solving Sin(^6)x + Cos(^6)x to solve other trigonometric expressions?

Yes, the formula for solving Sin(^6)x + Cos(^6)x can be used to solve other similar trigonometric expressions by recognizing and applying appropriate trigonometric identities. However, the process may differ depending on the specific expression and may require additional steps.

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