The value of ^210^ - 10^50^+^250^ is:

Question

The value of $\cos^210^\circ - \cos10^\circ\cos50^\circ+\cos^250^\circ$ is:

Accepted Answer

$$\frac{1+\cos20^\circ}2+\frac{1+\cos100^\circ}2 - \frac12(2\cos10^\circ\cos50^\circ) = 1+\frac12(\cos20^\circ+\cos100^\circ) - \frac12(\cos60^\circ+\cos40^\circ)$$

$$= 1-\frac14+\frac12[\cos20^\circ+\cos100^\circ-\cos40^\circ] = \frac34+\frac12[2\cos60^\circ\cos40^\circ - \cos40^\circ] = \frac34$$

(since $2\cos60^\circ\cos40^\circ=\cos40^\circ$, the bracket vanishes).

Solutions · 1