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lfdahl
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Find all possible $\theta$, that satisfy the equation:
$$\cos^8\theta+\sin^8\theta-2(1-\cos^2\theta\sin^2\theta)^2+1 = 0$$
$$\cos^8\theta+\sin^8\theta-2(1-\cos^2\theta\sin^2\theta)^2+1 = 0$$
[sp]lfdahl said:Find all possible $\theta$, that satisfy the equation:
$$\cos^8\theta+\sin^8\theta-2(1-\cos^2\theta\sin^2\theta)^2+1 = 0$$
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 $\theta=\dfrac{n\pi}{2}$, $n\in\mathbb{Z}$.
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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 $(1 - x(1-x))$.]
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This equation is used to find the solutions for a trigonometric expression, specifically involving the cosine and sine functions.
The equation involves trigonometric functions, so the degree cannot be determined as it depends on the value of θ.
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.
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.
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.