Variance of sine and cosine of a random variable

What is below is for $\mu=0$ (and variance renamed $\sigma^2$). Then $\mathbb{E}[\sin X]=0$, and you have $$ \operatorname{Var} \sin X = \mathbb{E}[\sin^2 X] = \frac{1}{2}\left(1-\mathbb{E}[\cos 2X]\right) $$ and $$ \mathbb{E}[\cos 2X] = \sum_{k=0}^\infty (-1)^k\frac{2^{2k}}{(2k)!} \mathbb{E}[X^{2k}] = \sum_{k=0}^\infty (-1)^k\frac{2^{2k}}{(2k)!} \sigma^{2k} (2k-1)!! = \sum_{k=0}^\infty (-1)^k \frac{2^{k}\sigma^{2k}}{k!} = e^{-2\sigma^{2}} $$ and therefore $$ \operatorname{Var} \sin X = \boxed{\frac{1-e^{-2\sigma^2}}{2}} $$ You can deal with the variance of $\cos X$ in a similar fashion (but you now have to substract a non-zero $\mathbb{E}[\cos X]^2$), especially recalling that $\mathbb{E}[\cos^2 X] = 1- \mathbb{E}[\sin^2 X]$.

Now, for non-zero mean $\mu$, you have $$ \sin(X-\mu) = \sin X\cos \mu - \cos X\sin\mu $$ (and similarly for $\cos(X-\mu)$) Since $X-\mu$ is a zero-mean Gaussian with variance $\sigma^2$, we have computed the mean and variance of $\sin(X-\mu)$, $\cos(X-\mu)$ already. You can use this with the above trigonometric identities to find those of $\cos X$ and $\sin X$. (it's a bit cumbersome, but not too hard.)

Without knowing anything about the distribution of $X$, I don't think there's much you can do.

Here is a general formulation using the law of the unconscious statistician that can be applied to other functions too. For specific calculations with $\sin$ and $\cos$ here though, I would say Clement C.'s answer is better!

The mean of $\color{blue}{h(X)}$ (for some function $h$) would be given by the integral $$\mathbb{E}[h(X)]=\int_{-\infty}^{\infty}\color{blue}{h(x)}f_X(x)\, dx,$$ where $f_X$ is the probability density function of $X$.

The second moment would be found similarly as $$\mathbb{E}\left[(h(X))^2\right] = \int_{-\infty}^{\infty}\color{blue}{(h(x)^2)}f_X(x)\, dx.$$

Once you know the first two moments here, you can calculate the variance using $\mathrm{Var}(Z) = \mathbb{E}[Z^2] - (\mathbb{E}[Z])^2$.

Replace $h(x)$ with $\cos x$ for the corresponding expectations for $\cos X$, and similarly with $\sin x$.

If the distribution of $X$ is not known, we cannot generally compute the exact mean and variance of $h(X)$. However, you may want to see this for some approximations that could be used. Some useful ones for you may be that if $X$ has mean $\mu_X$ and variance $\sigma^2_X$, then $$\mathbb{E}[h(X)]\approx h(\mu_X) + \dfrac{h''(\mu_X)}{2}\sigma_X^2$$ and $$\mathrm{Var}(h(X))\approx (h'(\mu_X)^2)\sigma^2_X + \dfrac{1}{2}(h''(\mu_X))^2 \sigma^4_X.$$