Proving uniqueness of solutions to $\sin^2A + \sin^2B = \sin (A+B)$ without using multivariable calculus

Question

In the course of solving a trigonometric problem (see $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle), in one approach the following equation needed to be solved:

$$\boxed{\sin^2A + \sin^2B = \sin (A+B)}$$

subject to $A\in(0,\frac{\pi}{2}),B\in(0,\frac{\pi}{2}),\pi-(A+B) > \max(A,B)$, i.e. $A$ and $B$ are the two angles of a triangle not opposite the longest side.

Clearly, any $A,B\:|\:A+B=\frac{\pi}{2}$ is a family of solutions. Since multivariable calculus is presumably beyond the level of the original problem:

How to prove that there are no other solutions $\underline{\text{without}}$ using multivariable calculus?

[I don't think a trig identity will suffice as there are other solutions if the restrictions on $A,B$ are relaxed.]

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(For completeness - using multivariable calculus)

Part 1

Proof that $\sin^2A + \sin^2B < \sin (A+B)$ over region $R_1=\{0<A,B\land A+B<\frac{\pi}{2}\}$.

Consider $$f(x,y)=\sin^2x + \sin^2y - \sin (x+y)$$ Then $$\begin{align} f_x &= \sin 2x - \cos(x+y) \\ f_y &= \sin 2y - \cos(x+y) \\ f_{xx} &= 2\cos 2x + \sin(x+y) \\ f_{yy} &= 2\cos 2y + \sin(x+y) \\ f_{xy} &= \sin(x+y) &= f_{yx} \\ \end{align}$$

For local extrema we require $f_x=f_y=0$. But

$$f_x=f_y=0 \implies \sin2x=\sin2y \implies y=x \lor y=\frac{\pi}{2}-x$$

Exclude $y=\frac{\pi}{2}-x$ as it violates the restriction that $x+y<\frac{\pi}{2}$.

If $y=x$, $f_x=0 \implies \sin2x=\sin2y \implies x=y=\frac{\pi}{8}$.

The determinant of the Hessian is $$D(x,y) = \begin{vmatrix} f_{xx} & f_{xy}\\ f_{yx} & f_{yy} \end{vmatrix} = f_{xx}f_{yy} - f_{xy}f_{yx} = f_{xx}f_{yy} - (f_{xy})^2$$

At $P(\frac{\pi}{8},\frac{\pi}{8})$, we have

$$\begin{align} f_{xx}&=2\cos\frac{\pi}{4}+\sin\frac{\pi}{4}=\frac{3}{\sqrt2} \\ f_{yy}&=2\cos\frac{\pi}{4}+\sin\frac{\pi}{4}=\frac{3}{\sqrt2} \\ f_{xy}&=\sin\frac{\pi}{4}=\frac{1}{\sqrt2} \\ D(x,y)&=\frac{9}{2}-\frac{1}{2}=4 \end{align}$$

Since both $f_{xx}$ and $D$ are positive at $P$, this is a local minimum (with $f(x,y)|_P=1-\sqrt2$).

On the boundaries:

• $f(0,y)=\sin^2y-\sin y<0$ on $x=0,y\in(0,\frac{\pi}{2})$
• $f(x,0)=\sin^2x-\sin x<0$ on $x\in(0,\frac{\pi}{2}),y=0$
• $f(0,0)=0$
• $f(x,y)=0$ for $x,y\geq0,x+y=\frac{\pi}{2}$

Since $(0,0)$ is not part of the domain, and there are no other local extrema, we have $f(x,y)<0$ over $R_1$.

Part 2

It will be sufficient to prove that $\sin^2A + \sin^2B > \sin (A+B)$ over region $R_2=\{0<A,B<\frac{\pi}{2} \land A+B>\frac{\pi}{2} \}$.

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Answer 1

Here are some illustrations, with $\triangle ABC$ having circumdiameter $1$ (and therefore sides $\sin A$, $\sin B$, $\sin C$):

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Acute $C$: $\quad \sin C < \sin^2 A + \sin^2 B$

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Obtuse $C$: $\quad \sin C > \sin^2 A + \sin^2 B$

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Right $C$: $\quad \sin C = \sin^2 A + \sin^2 B$

Answer 2

Well, fix $B$ and show that there's a unique solution for $A$. In other words, we want to show that for $0 < B < \pi/2$ the function

$$f_B(A) = \sin^2(A) + \sin^2(B) - \sin(A+B)$$

has a unique zero on the interval $(0, \pi/2)$.

First, note that $$f_B(0) = \sin^2(B) - \sin(B) = \sin(B)[\sin(B) - 1] < 0$$ since $0 < \sin(B) < 1$. Now consider

\begin{align*} f'_B(A) &= 2\sin(A) \cos(A) - \cos(A+B) \\ &= \sin(2A) - \cos(A+B) \end{align*} We have $$f'(0) = - \cos(B) < 0$$ so $f_B$ starts out negative and decreasing. Now, where can $f'_B$ vanish? Well, that happens when $$\sin(2A) = \cos(A+B).$$ Now $\sin(2A)$ increases from zero to one on $(0, \frac{\pi}{4})$ while $\cos(A+B)$ decreases from $\cos(B) > 0$ to $\cos(B + \frac{\pi}{4})$, which is between 0 and 1, on the same interval. Consequently there is some solution to this equation with $A \in (0, \pi/4)$. Moreover, there is only one solution on $(0, \pi/4)$, since to the right of this solution we have $\sin(2A)$ increasing and $\cos(A+B)$ decreasing.

What about solutions to $$\sin(2A) = \cos(A+B).$$ when $\frac{\pi}{4} < A < \frac{\pi}{2}$? Well, $$\cos(A+B) < \cos(A) < \sin(2A)$$ on this interval.

So we've shown that on the interval $(0, \pi/2)$ the function $f'_B(A)$ has a unique zero. Together with the facts that $f_B(0) < 0$ and $f'_B(0) < 0$ this means that on $(0, \pi/2)$ the function $f_B$ decreases from a negative value at 0 to a negative minimum, and then increases for the rest of the interval. Consequently, $f_B$ can have at most one zero on this interval.

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I suspect if you also object to single-variable calculus you could eliminate it from the argument and do things entirely with trigonometry, although the argument would necessarily be longer.

Answer 3

Suppose $a,b\in(0,\frac\pi2)$ and \begin{align*} \sin^2 a + \sin^2 b &= \sin(a+b) \\ &= \sin a\cos b + \cos a \sin b \tag{1} \end{align*} Squaring both sides, we get \begin{align*} \sin^4 a + 2\sin^2 a\sin^2 b + \sin^4 b &= \sin^2 a \cos^2 b + 2\sin a\sin b\cos a\cos b + \cos^2 a\sin^2 b \\ &= \sin^2 a (1-\sin^2 b) + 2\sin a\sin b\cos a\cos b + (1-\sin^2 a)\sin^2 b \\ &= \sin^2 a - 2\sin^2 a\sin^2 b + \sin^2 b + 2\sin a\sin b\cos a\cos b \end{align*} Rearranging, \begin{align*} 4\sin^2 a\sin^2 b &= \sin^2 a - \sin^4 a + \sin^2 b - \sin^4 b + 2\sin a\sin b\cos a\cos b \\ &= \sin^2 a(1-\sin^2 a) + \sin^2 b(1-\sin^2 b) + 2\sin a\sin b\cos a\cos b \\ &= \sin^2 a\cos^2 a + \sin^2 b\cos^2 b + 2\sin a\sin b\cos a\cos b \\ &= (\sin a\cos a + \sin b\cos b)^2 \end{align*} By our assumptions, everything is positive, so we can take square roots, getting $$ 2\sin a\sin b = \sin a\cos a + \sin b\cos b $$ Adding this to (1) yields $$ \sin^2 a + 2\sin a\sin b + \sin^2 b = \sin a\cos a + \sin a\cos b + \sin b\cos b + \sin b\cos a $$ that is, $$ (\sin a + \sin b)^2 = (\sin a + \sin b)(\cos a + \cos b) $$ Therefore (again, everything being positive) $$ \sin a + \sin b = \cos a + \cos b $$ Rearranging, $$ \sin a - \cos a = \cos b - \sin b $$ and squaring, $$ \sin^2 a - 2\sin a\cos a + \cos^2 a = \sin^2 b - 2\sin b\cos b + \cos^2 b $$ Simplifying, $$ \sin 2a = \sin 2b $$ Since $a,b\in(0,\frac\pi2)$, we conclude that either $a=b$ or $a+b=\frac\pi2$. In the case $a=b$, (1) reduces to $\sin 2a = 2\sin^2 a = 1 - \cos 2a$, which is equivalent to $\sin 4a = 0$, giving (in our interval) only the solution $a=\frac\pi4$, which is also part of the second case. So $a+b=\frac\pi2$.