The number of solutions of the equation 1+^4x=^23x, x[-52,52] is:

Question

The number of solutions of the equation $1+\sin^4x=\cos^23x$, $x\in\left[-\dfrac{5\pi}2,\dfrac{5\pi}2\right]$ is:

Accepted Answer

Since $1+\sin^4x \ge 1$ and $\cos^23x \le 1$, equality requires both $\sin^4x=0$ and $\cos^23x=1$:

$$\sin x = 0 \implies x=n\pi$$

At $x=n\pi$: $\cos3x=\cos(3n\pi)=(-1)^{3n}=\pm1$, so $\cos^23x=1$ automatically. So every $x=n\pi$ works.

Within $\left[-\tfrac{5\pi}2,\tfrac{5\pi}2\right]$: $x=-2\pi,-\pi,0,\pi,2\pi$ --- 5 solutions.

Solutions · 1