1961 IMO Problems/Problem 3

Problem

Solve the equation

$\cos^n{x} - \sin^n{x} = 1$

where $n$ is a given positive integer.

Solution

Since $cos^2x + sin^2x = 1$ , we cannot have solutions with $n\ne2$ and $0<|cos(x)|,|sin(x)|<1$ . Nor can we have solutions with $n=2$ , because the sign is wrong. So the only solutions have $sin (x) = 0$ or $cos (x) = 0$ , and these are: $x =$ multiple of $\pi$ , and $n$ even; $x$ even multiple of $\pi$ and $n$ odd; $x$ = even multiple of $\pi + \frac{3\pi}{2}$ and $n$ odd.

1961 IMO (Problems) • Resources
Preceded by Problem 2	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 4
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1961 IMO Problems/Problem 3

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Solution