## Task 25/76

Solve the equation $\sin ^{4} x+\cos ^{7} x=1$ for real numbers $x$.
Due to

$$
\sin ^{4} x=\left(\sin ^{2} x\right)^{2}=\left(1-\cos ^{2} x\right)^{2}
$$

it follows from the equation after a short calculation

$$
\sin ^{4} x+\cos ^{7} x=1 \rightarrow \cos ^{2} x\left(\cos ^{5} x+\cos ^{2} x-2\right)=0
$$

1. This equation is satisfied for $\cos x=0, x=\frac{\pi}{2}+k \pi, \sin x= \pm 1$. The verification confirms the correctness.
2. This equation is satisfied for $\cos ^{5} x+\cos ^{2} x-2=0$. Since $\cos x \leq 1, \cos ^{5} x \leq 1$, this equation can only be satisfied for $\cos x=1, x=2 k \pi, \sin x=0$. The verification confirms the correctness.