Solve $\sin^{3}x+\cos^{3}x=1$

Solution 1:

Hint $$\sin^3(x)+\cos^3(x)=\sin^2(x)+\cos^2(x).$$

Solution 2:

Both of $\sin(x),\cos(x)$ must be nonnegative since, if one of them was negative, the equation $\sin^3(x)+\cos^3(x)=1$ would imply that the other one is more than $1$, contradiction.

Thus, we have $0\le \sin(x) \le 1$ and $0\le \cos(x)\le 1$.

If both of $\sin(x),\cos(x)$ are less than $1$, then, since they are both nonnegative, we would have $$ \begin{cases} 0\le \sin^3(x) < \sin^2(x)\\[4pt] 0\le \cos^3(x) < \cos^2(x)\\ \end{cases} $$ but that would imply $$\sin^3(x)+\cos^3(x) < \sin^2(x)+\cos^2(x)=1$$ contradiction.

It follows that one of $\sin(x),\cos(x)$ must be equal to $1$, and the other must be equal to zero.

From that information, I'm sure you can finish the solution.

Solution 3:

Hint:

Let $\sin x+\cos x=t\implies t^2=?$

$$1=\dfrac{t\{2-(t^2-1)\}}2\iff t^3-3t+2=0$$

Clearly, $t=1$ a solution

Solution 4:

Let use by $t= \tan (x/2)$

$\sin x=\frac{2t}{1+t^2}$
$\cos x=\frac{1-t^2}{1+t^2}$

to obtain

$$2t^6-8t^3+6t^2=0$$

$$\iff t^2(t^4-4t+3)=t^2(t-1)^2(t^2+2t+3)=0$$

and since $t^2+2t+3>0$ the solutions are

$t=0 \implies \frac x 2=k\pi\implies x=2k\pi$
$t=1 \implies \frac x 2=\frac{\pi}4+k\pi \implies x=\frac{\pi}2+2k\pi $

Solution 5:

As an alternative by

$$\sin^3 x+ \cos ^3 x=1 \iff \sin x\cdot \sin^2+\cos x\cdot \cos^2 x=1$$

since $\sin^2 x+ \cos ^2 x=1$, the given equality is a weighted mean of $\sin x$ and $\cos x$ which holds if and only if

$\sin x=1,\,\cos x=0$

$\cos x=1,\,\sin x=0$