Practice (115)
Let $(x^{2014} + x^{2016} +2)^{2015} = a_0 + a_1 x + \cdots + a_nx^n$. Find $$a_0 -\frac{a_1}{2} -\frac{a_2}{2} + a3 - \frac{a_4}{2}-\frac{a_5}{2} + a_6 - \cdots$$
Simplify $$\sin{x} + \sin{2x} + \cdots +\sin{nx}$$ and $$\cos{x} + \cos{2x} + \cdots + \cos{nx}$$
Solve the equation $\cos\theta + \cos 2\theta + \cos 3\theta = \sin \theta +\sin 2\theta + \sin 3\theta$.
Let $A, B,$ and $C$ be angles of a triangle. If $\cos 3A + \cos 3B + \cos 3C = 1$, determine the largest angle of the triangle.
Find the value of $\cos 20^\circ \cos 40^\circ \cos 80^\circ$ using at least two difference methods.
Simplofy $\sin\theta + \frac{1}{2}\cdot\sin 2\theta + \frac{1}{4}\cdot\sin 3\theta + \cdots$.
Solve the equation $\cos^2 x + \cos^2 2x +\cos^2 3x=1$ in $(0, 2\pi)$.
Prove for every positive integer $n$ and real number $x\ne \frac{k\pi}{2^t}$ where $t =0, 1, 2,\cdots$ and $k$ is an integer, the following relation always holds:
$$\frac{1}{\sin 2x}+\frac{1}{\sin 4x} + \cdots +\frac{1}{\sin 2^nx}=\frac{1}{\tan x}-\frac{1}{\tan 2^nx}$$
Let $(x^{2014} + x^{2016}+2)^{2015}=a_0 + a_1x+\cdots+a_nx^2$, Find the value of $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$
Prove the triple angle formulas: $$\sin 3\theta = 3\sin\theta -4\sin^3\theta$$ and $$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$$
Find the value of the following expression: $$\binom{2020}{0}-\binom{2020}{2}+\binom{2020}{4}-\cdots+\binom{2020}{2020}$$
Given integers $a$, $b$, $n$. Show that there exist integers $x$, $y$, such that $$(a^2+b^2)^n = x^2 + y^2$$.
The points $(0,0)$, $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
Let $n$, $r$, and $m$ all be positive integers, $r\le m$, and $\omega_k=e^{\frac{2k\pi}{m}i}$ be a complex root to the equation $x^m=1$. Show $$\sum_{k=0}^{\lfloor{\frac{n-r}{m}}\rfloor}\binom{n}{r+km}x^{r+km}=\frac{1}{m}\sum_{k=0}^{m-1}\omega^{-r}(1+x\omega_k)^n$$
where function $\lfloor{x}\rfloor$ returns the largest integer not exceeding real number $x$.