MHB Cos^2(∠POA)+cos^2(∠POB)+cos^2(∠POC)+cos^2(∠POD) is independent of P.

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The expression \(\cos^2(\angle POA) + \cos^2(\angle POB) + \cos^2(\angle POC) + \cos^2(\angle POD)\) is shown to be independent of the point \(P\) on the unit sphere. By selecting specific vertices of the inscribed cube and calculating the cosine values based on the coordinates of \(P\), it is demonstrated that the resulting sum simplifies to a constant value of \(\frac{4}{3}\). This result holds regardless of the position of \(P\) on the sphere, confirming the independence of the expression from \(P\). The calculations utilize the properties of dot products and the geometry of the cube and sphere. Ultimately, the discussion reinforces the mathematical relationship between the angles and the fixed geometric structure.
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A cube is inscribed in the unit sphere $x^2 +y^2 +z^2 = 1$. Let $A,B,C$ and $D$ denote the
vertices of one face of the cube. Let $O$ denote the center of the sphere, and $P$
denote a point on the sphere. Show that

\[\cos ^2(\angle POA)+\cos ^2(\angle POB)+\cos ^2(\angle POC)+\cos ^2(\angle POD)\]

is independent of $P$.
 
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lfdahl said:
A cube is inscribed in the unit sphere $x^2 +y^2 +z^2 = 1$. Let $A,B,C$ and $D$ denote the
vertices of one face of the cube. Let $O$ denote the center of the sphere, and $P$
denote a point on the sphere. Show that

\[\cos ^2(\angle POA)+\cos ^2(\angle POB)+\cos ^2(\angle POC)+\cos ^2(\angle POD)\]

is independent of $P$.
[sp]
Let us write $S$ for the expression, and $R$ for the radius of the sphere. As $S$ is dimensionless, it is independent of $R$, and we may choose $R$ as we want.

We take the vertices as $A=(1,1,1)$, $B=(-1,1,1)$, $C=(1,-1,1)$ and $D=(-1,-1,1)$; this makes $R=\sqrt3$.

Let the coordinates of $P$ be $(x,y,z)$. We have:

$\cos POA = \dfrac{\langle OA\cdot OP\rangle}{R^2} = \dfrac{x + y + z}{3}$

and a similar expression for the other terms ($\langle\cdot\rangle$ is the dot product).

This gives:

$$\begin{align*}
S &=\frac19\left((x+y+z)^2 + (-x+y+z)^2 + (x-y+z)^2 +(-x-y+z)^2\right)\\
&= \frac49\left(x^2+y^2+z^2\right)
\end{align*}
$$

and, as $x^2+y^2+z^2=R^2 = 3$, we get $S=\dfrac43$, which is indeed independent of the position of $P$.
[/sp]
 
castor28 said:
[sp]
Let us write $S$ for the expression, and $R$ for the radius of the sphere. As $S$ is dimensionless, it is independent of $R$, and we may choose $R$ as we want.

We take the vertices as $A=(1,1,1)$, $B=(-1,1,1)$, $C=(1,-1,1)$ and $D=(-1,-1,1)$; this makes $R=\sqrt3$.

Let the coordinates of $P$ be $(x,y,z)$. We have:

$\cos POA = \dfrac{\langle OA\cdot OP\rangle}{R^2} = \dfrac{x + y + z}{3}$

and a similar expression for the other terms ($\langle\cdot\rangle$ is the dot procuct).

This gives:

$$\begin{align*}
S &=\frac19\left((x+y+z)^2 + (-x+y+z)^2 + (x-y+z)^2 +(-x-y+z)^2\right)\\
&= \frac49\left(x^2+y^2+z^2\right)
\end{align*}
$$

and, as $x^2+y^2+z^2=R^2 = 3$, we get $S=\dfrac43$, which is indeed independent of the position of $P$.
[/sp]

Thankyou, castor28 for such a nice solution and for your participation!(Handshake)
 
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