Can $\sin^5 x + \cos^3 x$ Equal 1?

[anemone]

Here is this week's POTW:

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Find all the real solutions of the equation $\sin^5 x+\cos^3 x=1$.

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[anemone]

Congratulations to the following members for their correct solution!(Cool)

1. Opalg
2. kaliprasad
3. lfdahl

Solution from Opalg:

Spoiler

Let $f(x) = \sin^5x+ \cos^3x - 1$. Since $f$ has period $2\pi$ it will be enough to find solutions of $f(x)=0$ in the interval $0\leqslant x < 2\pi$.

If $\sin x$ or $\cos x$ is negative then $f(x)<0$. That rules out the interval $\pi/2 <x < 2\pi$. So it will be enough to find solutions of $f(x)=0$ in the interval $0\leqslant x \leqslant \pi/2$.

If $x=0$ or $x=\pi/2$ then $f(x)=0$. That leaves the interval $0<x<\pi/2$. But in that interval $0<\sin x <1$ and $0<\cos x <1$. Therefore $ \sin^5x+ \cos^3x < \sin^2x+ \cos^2x =1$ and so $f(x) <0$.

Thus the only solutions are $x = 2k\pi$ and $x = \left(2k+\frac12\right)\pi$ for $k\in\Bbb{Z}$.