Solve $\cos x +\cos y - \cos(x+y)=\frac 3 2$

Question

Solve $\cos x +\cos y - \cos(x+y)=\frac 3 2$ where $x,y\in [0,\pi]$.

I am trying to solve this but I am stuck. I know that $x=y=\pi/3$ is a solution but how do I show this is the only one? I think there are no others! Hints would be appreciated

Answer 1

For $x,y \in [0,\pi]$

\begin{align} \cos x &= u \\ \cos y &= v \\ \sin x &= \sqrt{1-u^2} \\ & \ge 0 \\ \sin y &=\sqrt{1-v^2} \\ & \ge 0 \\ u+v-uv+\sqrt{1-u^2} \sqrt{1-v^2} &= \frac{3}{2} \\ \sqrt{1-u^2} \sqrt{1-v^2} &= \frac{3}{2}+uv-u-v \end{align}

By AM $\ge$ GM,

$$\frac{1-u^2+1-v^2}{2} \ge \frac{3}{2}+uv-u-v \tag{1}$$

equality holds if $1-u^2=1-v^2$ and rearranging $(1)$ gives

$$0 \ge \frac{(u+v-1)^2}{2} \tag{2}$$

But $\dfrac{(u+v-1)^2}{2}$ is non-negative which forces the equality to hold.

Therefore,

$$u+v=1 \land u^2=v^2$$

$$u+v=1 \land (u+v)(u-v)=0$$

$$u=v=\frac{1}{2} \implies x=y=\frac{\pi}{3}$$

Further points to be noticed

Considering the quadratic in $u$ of

$$(1-u^2)(1-v^2)=\left( \frac{3}{2}+uv-u-v \right)^2$$

Discriminant $$\Delta_u =4v^4-4v^3-3v^2+4v-1=(v^2-1)(2v-1)^2$$

• $v^2>1 \implies \Delta_u>0$ but $v$ is beyond the range of cosine

• $v=1 \implies \Delta_u=0$ but gives $0=\frac{1}{4}$, so no solutions

• $v=-1 \implies \Delta_u=0$ but gives $u=\frac{5}{4}$ which is beyond the range of cosine

• $v=\frac{1}{2} \implies \Delta_u=0$ that gives $u=\frac{1}{2}$

• $\Delta_u<0$ elsewhere

Answer 2

It's a classic "quadratic equation" problem... Put $t = \cos(\frac{x+y}{2})\implies 2t\cos(\frac{x-y}{2})-2t^2+1=\dfrac{3}{2}\implies 4t^2-4t\cos(\frac{x-y}{2})+1=0$. $\triangle'=(b')^2-ac= 4\cos^2(\frac{x-y}{2})-4 \le 0\implies \triangle' = 0\implies \cos(\frac{x-y}{2}) = \pm 1, (2t\mp 1)^2=0\implies t = \pm \frac{1}{2}\implies x = y = \frac{\pi}{3}$

Answer 3

\begin{align} \cos x +\cos y - \cos(x+y) &= \frac 32 \\ \cos x + \cos y - \cos x \cos y + \sin x \sin y &= \frac 32 \\ (1 - \cos y)\cos x + \sin y \sin x &= \frac 32 - \cos y \\ \end{align}

Note that $$\sqrt{(1-\cos y)^2 + (\sin y)^2} = \sqrt{1 - 2\cos y + \cos^2y + \sin^2 y} = \sqrt{2(1 - \cos y)} = 2\sin \dfrac y2$$

$$\dfrac{1-\cos y}{2 \sin \dfrac y2} = \sin \dfrac y2$$

$$ \dfrac{\sin y}{2 \sin \dfrac y2} = \cos \dfrac y2$$

So \begin{align} \sin \dfrac y2\cos x + \cos \dfrac y2 \sin x &= \frac{\frac 32 - \cos y}{2 \sin \dfrac y2} \\ \sin \left(x + \dfrac y2 \right) &= \frac{3 - 2\cos y}{4 \sin \dfrac y2} \\ \sin \left(x + \dfrac y2 \right) &= \frac{1 + 4\sin^2 \dfrac y2}{4 \sin \dfrac y2} \\ \sin \left(x + \dfrac y2 \right) &= \sin \dfrac y2 + \frac 14 \csc \dfrac y2\\ \end{align}

We can show that, for $y \in [0, \pi]$, the minimum acceptable value of $\sin \dfrac y2 + \frac 14 \csc \dfrac y2$ is $1$. The rest is pretty straight forward.