Solve cospi/n

$$\cos \frac { \pi } { n }$$

Differentiate w.r.t. n

$$\frac{\pi \sin(\frac{\pi }{n})}{n^{2}}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\left(-\sin(\pi \times \frac{1}{n})\right)\frac{\mathrm{d}}{\mathrm{d}n}(\pi \times \frac{1}{n})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\left(-\sin(\pi \times \frac{1}{n})\right)\left(-1\right)\pi n^{-1-1}$$

Simplify.

$$\left(-\left(-\frac{\pi }{n^{2}}\right)\right)\sin(\pi \times \frac{1}{n})$$

Evaluate

$$\cos(\frac{\pi }{n})$$