Let $x$, $y$, $z$ be three positive real numbers satisfying $xyz+x+z=y$. Find the maximum value of $$P=\frac{2}{x^2 + 1}-\frac{2}{y^2+1}+\frac{3}{z^2+1}$$
Let real numbers $x$ and $y$ satisfy the relation $4x^2-5xy+4y^2=5$. Find the maximum and minimal value of $x^2+y^2$.
Given non-negative real numbers $x$, $y$ and $z$, prove
$$\sqrt{x^2+y^2-xy}+\sqrt{y^2 + z^2 - yz}\ge\sqrt{x^2+z^2+xz}$$
Given any five real numbers, show that at least two of them $x$ and $y$ satisfy the condition $|xy+1|>|x-y|$.
Let $\{x_n\}$ and $\{y_n\}$ be two real number sequences which are defined as follow:
$$x_1=y_1=\sqrt{3},\quad x_{n+1}=x_n +\sqrt{1+x_n^2},\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$
for all $n\ge 1$. Prove that $2 < x_ny_n < 3$ for all $n>1$.
Let $m$ be a positive integer. Show that $$\frac{1}{\sqrt{m+1}}< \sin\frac{1}{\sqrt{m}}$$
Let $x\in(0, \pi/2)$ be expressed in radian. Explain why the relation $\sin x < x < \tan x$ hold?
Let $m$ be a positive integer. Show that $$\sin\frac{2}{\sqrt{m}}< \frac{2}{\sqrt{m+1}}$$
Prove $$\frac{1}{\sqrt{2019}} < \underbrace{\sin\sin\sin\cdots\sin}_{2017}\frac{\sqrt{2}}{2} < \frac{2}{\sqrt{2019}}$$
Solve this equation $$2\sqrt{2}x^2 + x -\sqrt{1-x ^2}-\sqrt{2}=0$$
If $\triangle{ABC}$ is not a right triangle, show
\begin{equation}
\tan A + \tan B + \tan C =\tan A \tan B \tan C
\end{equation}
In $\triangle{ABC}$, show that $$\cot A\cot B +\cot B\cot C + \cot C\cot A = 1$$
Given $\triangle{ABC}$, show that
$$\cos A + \cos B + \cos C =1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$$
Let $\theta\in[0, 2\pi]$ satisfying $$\cos^5\theta -\sin^5\theta < 7(\sin^3\theta -\cos^3\theta)$$
Find the range of $\theta$.
In $\triangle{ABC}$, show that
\begin{align*}
\sin 2A + \sin 2B + \sin 2C &= 4\sin A\sin B \sin C\\
\cos 2A + \cos 2B + \cos 2C &= -1-4\cos A\cos B\cos C
\end{align*}
In $\triangle{ABC}$, show that
\begin{align*}
&\sin^2A +\sin^2B+\sin^2C = 2 +2\cos A\cos B \cos C\\
&\cos^2A +\cos^2B + \cos^2C = 1-2\cos A\cos B\cos C
\end{align*}
Compute the values of
$$S=C_n^1\sin\theta + C_n^2\sin 2\theta + \cdots + C_n^n\sin n\theta$$
and
$$C=C_n^1\cos\theta + C_n^2\cos 2\theta + \cdots + C_n^n\cos n\theta$$
$$|\sin x + \cos x + \tan x + \cot x + \sec x + \csc x|$$
where $x$ is a real number.
Let $a$ and $b$ be two positive real numbers not exceeding $1$. Prove $$\frac{1}{\sqrt{a^2 + 1}}+\frac{1}{\sqrt{b^2 +1}}\le\frac{2}{\sqrt{1+ab}}$$
Solve this inequality $$\frac{x}{\sqrt{x^2 +1}}+\frac{1-x^2}{1+x^2} > 0$$
Compute $\sin 15^\circ$ and $\cos 15^\circ$ using a geometry approach.
Find the range of real number $a$ if the following equation of $x$ is solvable in real number:
$$\sin^2 x + \cos x + a=0$$
Find the number of solutions to the equation $\sin x = \frac{x}{2018}$.
Prove that function $f(x)=\cos\sqrt{x}$ is not a periodical function.
Sort $\sin(-1)$, $\cos(-1)$, and $\tan(-1)$ in an ascending order.