To solve the problem, we start by using trigonometric identities to simplify the given equation.

Step 1: Apply trigonometric identities

Recall the co-function identities:

• $\sin\left(\frac{\pi}{2} - x\right) = \cos x$
• $\cos\left(\frac{\pi}{2} + x\right) = -\sin x$

Step 2: Substitute the identities into the equation

The original equation is:
$2\sin\left(\frac{\pi}{2} - x\right) + 3\cos\left(\frac{\pi}{2} + x\right) = 0$

Substitute the identities:
$2\cos x + 3(-\sin x) = 0$

Step 3: Rearrange to find $\tan x$

Simplify the equation:
$2\cos x - 3\sin x = 0$

Rearrange:
$2\cos x = 3\sin x$

Divide both sides by $\cos x$ (since $\cos x \neq 0$ here):
$2 = 3\tan x$

Thus:
$\tan x = \frac{2}{3}$

Answer: $\boxed{\dfrac{2}{3}}$

作者声明:本文包含人工智能生成内容。