Show that $\binom{n}{0}+\binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{n} = 2^n$.
Explain that $$\sin x +\sin (x+120^\circ) + \sin (x-120^\circ) = \cos x +\cos (x+120^\circ) + \cos (x-120^\circ) = 0$$ using at least two approaches.
(Sophie Germain's Identity) Prove $a^4 + 4b^4 = (a^2+2b^2-2ab)(a^2+2b^2+2ab)$.
(Bezout's theorem) Show that two positive integers $a$ and $b$ are co-prime if there exist integer $x$ and $y$ satisfying $ax+by=1$.
(Apollonius’ Theorem) Let $AD$ be one median of $\triangle{ABC}$ where point $D$ lies on side $BC$. Show that the following relation holds:
$$AB^2 +AC^2 = 2\times(AD^2 +BD^2)$$
If $0 < \alpha < \beta < \frac{\pi}{2}$, show $$\frac{\cot\beta}{\cot\alpha}<\frac{\cos\beta}{\cos\alpha}<\frac{\beta}{\alpha}$$
Given any $\triangle{ABC}$, show that $$\cos A + \cos B + \cos C = 1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$$
Show that $$S_{\triangle{ABC}}=\frac{abc}{4R}$$
Find the value of $$\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
Show that
$$\sec^2\alpha = 1 + \tan^2\alpha$$
$$\csc^2\alpha = 1 + \cot^2\alpha$$
Prove the identity: $\tan^2 x - \sin^2 x = \tan^2 x \sin^2 x$.
If $\cos x - \sin x = \sqrt{2}\sin x$, prove $\cos x +\sin x = \sqrt{2}\cos x$.
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
$$\cos A + \cos B + \cos C =1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$$
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*}
Sort $\sin(-1)$, $\cos(-1)$, and $\tan(-1)$ in an ascending order.
Prove the following identity
\begin{equation}
\tan\alpha + \tan(90^\circ - \alpha)=\frac{2}{\sin 2\alpha}
\end{equation}
Compute the value of $(\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ)$.
Compute $\sin 25^\circ \sin 35^\circ \sin 85^\circ$.
(Pascal Identity) Show $\binom{n}{k} + \binom{n}{k+1}=\binom{n+1}{k+1}$.
John uses the equation method to evaluate the following expression:$$S=1-1+1-1+1-\cdots$$ and get $$S=1-S \implies \boxed{S=\frac{1}{2}}$$
However, $S$ clearly cannot be a fraction. Can you point out what is wrong here?
Let real numbers $a$ and $b$ satisfy $0 < a < a +\frac{1}{2} \le b$ and $a^{40}+b^{40}=1$. Show that all the twelve digits after the decimal point are $9$ if $b$ is expressed in decimal.
Let $z=\cos{\theta} + i\sin{\theta} $. Show $z^{-1} = \cos{\theta} - i\sin{\theta}$.