Solve the equation cos^8θ+sin^8θ−2(1−cos^2θsin^2θ)^2+1=0

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In summary, the conversation discusses finding all possible values of $\theta$ that satisfy the given equation involving $\cos$ and $\sin$ functions. By substituting $x=\cos^2\theta$ and simplifying, it is shown that the equation is a tautology and therefore true for all values of $\theta$.
  • #1
lfdahl
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Find all possible $\theta$, that satisfy the equation:

$$\cos^8\theta+\sin^8\theta-2(1-\cos^2\theta\sin^2\theta)^2+1 = 0$$
 
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  • #2
lfdahl said:
Find all possible $\theta$, that satisfy the equation:

$$\cos^8\theta+\sin^8\theta-2(1-\cos^2\theta\sin^2\theta)^2+1 = 0$$
[sp]
Let us write $x=\cos^2\theta$; this implies $\sin^2\theta=1-x$. The equation becomes
$$\begin{align*}
x^4 + (1-x)^4 -2(1 - x(1-x))+1&=0\\
2x^4-4x^3+4x^2-2x&=0\\
2x(x-1)(x^2-x+1)&=0
\end{align*}$$
As $x^2-x+1$ has no real root, the only solutions are $x=0$ and $x=1$ (since $x\ge0$). These correspond to $\cos\theta=0,\,\pm1$ and $\theta=\dfrac{n\pi}{2}$, $n\in\mathbb{Z}$.
[/sp]
 
  • #3
[sp]
$$\cos^8\theta+\sin^8\theta-2(1-\cos^2\theta\sin^2\theta)^2+1 = 0$$
$$\cos^8\theta+\sin^8\theta-2\cos^4\theta\sin^4\theta +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$(\cos^4\theta - \sin^4\theta)^2 +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$\bigl((\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)\bigr)^2 +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$(\cos^2\theta - \sin^2\theta)^2 +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$(\cos^2\theta + \sin^2\theta)^2 - 1 = 0$$
$$1 - 1 = 0$$ That is a tautology, so the equation is true for all $\theta$.

[It looks as though castor28 omitted to square $(1 - x(1-x))$.]
[/sp]
 
  • #4
castor28 said:
[sp]
Let us write $x=\cos^2\theta$; this implies $\sin^2\theta=1-x$. The equation becomes
$$\begin{align*}
x^4 + (1-x)^4 -2(1 - x(1-x))+1&=0\\
2x^4-4x^3+4x^2-2x&=0\\
2x(x-1)(x^2-x+1)&=0
\end{align*}$$
As $x^2-x+1$ has no real root, the only solutions are $x=0$ and $x=1$ (since $x\ge0$). These correspond to $\cos\theta=0,\,\pm1$ and $\theta=\dfrac{n\pi}{2}$, $n\in\mathbb{Z}$.
[/sp]

Thankyou, castor28, for your participation!

It seems, Opalg is right: You forgot to square the parenthesis. The correct answer is: $\theta \in \Bbb{R}$.
 
  • #5
Opalg said:
[sp]
$$\cos^8\theta+\sin^8\theta-2(1-\cos^2\theta\sin^2\theta)^2+1 = 0$$
$$\cos^8\theta+\sin^8\theta-2\cos^4\theta\sin^4\theta +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$(\cos^4\theta - \sin^4\theta)^2 +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$\bigl((\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)\bigr)^2 +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$(\cos^2\theta - \sin^2\theta)^2 +4\cos^2\theta\sin^2\theta - 1 = 0$$
$$(\cos^2\theta + \sin^2\theta)^2 - 1 = 0$$
$$1 - 1 = 0$$ That is a tautology, so the equation is true for all $\theta$.

[It looks as though castor28 omitted to square $(1 - x(1-x))$.]
[/sp]

Thankyou, Opalg, for a short and elegant solution!

Yes, the equation is indeed a tautology!
 

FAQ: Solve the equation cos^8θ+sin^8θ−2(1−cos^2θsin^2θ)^2+1=0

1. What is the equation cos^8θ+sin^8θ−2(1−cos^2θsin^2θ)^2+1=0 used for?

This equation is used to find the solutions for a trigonometric expression, specifically involving the cosine and sine functions.

2. What is the degree of the equation cos^8θ+sin^8θ−2(1−cos^2θsin^2θ)^2+1=0?

The equation involves trigonometric functions, so the degree cannot be determined as it depends on the value of θ.

3. How do I solve the equation cos^8θ+sin^8θ−2(1−cos^2θsin^2θ)^2+1=0?

To solve this equation, you can use algebraic manipulations and trigonometric identities to simplify it. Then, you can use the quadratic formula to find the roots of the simplified equation.

4. What are the possible values of θ that satisfy the equation cos^8θ+sin^8θ−2(1−cos^2θsin^2θ)^2+1=0?

The values of θ that satisfy the equation will depend on the solutions found after simplifying and solving the equation. Since the equation involves cosine and sine functions, the solutions will likely involve values of θ that make these functions equal to 0 or 1.

5. Can this equation be solved by hand or do I need to use a calculator?

This equation can be solved by hand using algebraic and trigonometric manipulation techniques. However, depending on the values of θ and the complexity of the equation, a calculator may be more efficient and accurate.

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