Answer to Question #350746 in Algebra for Biola

Question

Find x if, 2^sin²x + 2^cos²x = 3

Answer 1

$2^{\sin^2x}+2^{\cos^2x}=3$

$2^{\sin^2x}+2^{1-\sin^2x}=3$

$2^{\sin^2x}+2\cdot\frac{1}{2^{\sin^2x}}=3$

Let $y=2^{\sin^2x}$

$y+\frac{2}{y}=3$

$y^2-3y+2=0$

$y_1=1$, $y_2=2$

$2^{\sin^2x}=1$

$\sin^2x=0$

$\sin x=0$

$x=\pi n, n\in\mathbb{Z}$

$2^{\sin^2x}=2$

$\sin^2x=1$

$\sin x=\pm1$

$x=(-1)^n\frac{\pi}{2}+\pi n, n\in\mathbb{Z}$

Answer:

$x=\pi n, n\in\mathbb{Z}$

$x=(-1)^n\frac{\pi}{2}+\pi n, n\in\mathbb{Z}$