Please check my work.
Let $\mu$ be the uniform measure on $\mathbb{S}^n$, the unit sphere with radius $1$, and $$A=\{x\in\mathbb{S}^n:x_1^2+x_2^2\le \sin^2\alpha\}$$
where $\alpha$ is constant.
Claim $$\mu(A)=1-\cos^{n-1}(\alpha)$$
Solution
For any angle $\varphi$, the slice
$$\{x\in\mathbb{S}^n:x_1^2+x_2^2=\sin^2\varphi\}$$
is the product of a $1$-sphere with radius $\sin\varphi$ and an $n-2$ sphere with radius $\cos\varphi$. So, the measure of the slice $x_1^2+x_2^2=\sin^2\varphi$ is the product $$\left(\int_{\mathbb{S}^1} 1\right) \cdot c\sin\varphi\left(\int_{\mathbb{S}^{n-2}} 1\right)\cdot C \cos^{n-2}\varphi$$ for some constants $c$ and $C$. Then, uniform measure is the ratio of the measure of $A$ to the measure of the whole sphere:
$$\mu(A)=\frac{\int_0^\alpha (c\sin\varphi)(C\cos^{n-2}\varphi)d\theta}{\int_0^{\pi/2} (c\sin\varphi)(C\cos^{n-2}\varphi)d\theta}$$
The constants cancel and give the result.