4. Calculate $\cos (\pi+2 x)$, if $\cos x=\frac{1}{4}$.
4. We use the identity for the transition to the acute angle $\cos (\pi+2 x)=-\cos 2 x$, then the double angle identity $-\cos 2 x=-\cos ^{2} x+\sin ^{2} x$. We use the identity for $\sin ^{2} x=1-\cos ^{2} x$. We simplify the expression, substitute $\cos x=\frac{1}{4}$. We get $1-2 \cos ^{2} x=1-2 \cdot\left(\frac{1}{4}\right)^{2}$. We calculate the value of the expression $\frac{7}{8}$.

Using the transition to the acute angle $\cos (\pi+2 x)=-\cos 2 x \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Using the double angle identity $-\cos 2 x=-\cos ^{2} x+\sin ^{2} x \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=54&width=1636&top_left_y=2303&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=57&width=1636&top_left_y=2353&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=63&width=1639&top_left_y=2401&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_6d1b766ff5bddd57ba61g-13.jpg?height=60&width=1639&top_left_y=2460&top_left_x=206)