Practice (118)
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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.
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 $\sin 25^\circ \sin 35^\circ \sin 85^\circ$.
Prove $\tan 20^\circ \tan 40^\circ \tan 60^\circ \tan 80^\circ=3$.
Show that $\cot 70^\circ + 4\cos 70^\circ = \sqrt{3}$.
Prove that $\cos 1^\circ$ is irrational.
In $\triangle{ABC}$, find the measurement of $C$ if
$$3\sin A + 4\cos B = 6\quad\text{and}\quad 4\sin B + 3\cos A = 1$$
Compute the value of $(\sqrt{3}\tan 18^\circ + \tan 18^\circ\tan 12^\circ +\sqrt{3}\tan 12^\circ)$.
Compute the value of $\cos 6^\circ\cos 42^\circ \cos 66^\circ\cos 78^\circ$.
Show that for any positive integer: $$\tan x \tan 2x +\tan 2x \tan 3x +\cdots + \tan(n-1)x\tan nx=\frac{\tan nx}{\tan x}-n$$
Show that $$\tan x + 2\tan 2x + 2^2\tan 2^2x +\cdots + 2^n\tan 2^nx = \cot x - 2^{n+1}\cot 2^{n+1}x$$
Show that $$\frac{1}{\sin 1^\circ\sin 2^\circ}+\frac{1}{\sin 2^\circ\sin 3^\circ}+\cdots+\frac{1}{\sin 89^\circ\sin 90^\circ}=\cos 1^\circ\csc^2 1^\circ$$
As shown, both $ABCD$ and $OPRQ$ are squares. Additionally, $O$ is the center of $ABCD$, $OP=1$, $BP=\sqrt{2}$, and $CQ=\sqrt{5}$. Find the length of $DR$.
In $\triangle ABC, AB = AC = 10$ and $BC = 12$. Point $D$ lies strictly between $A$ and $B$ on $\overline{AB}$ and point $E$ lies strictly between $A$ and $C$ on $\overline{AC}$ so that $AD = DE = EC$. Then $AD$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle{ABC}$ respectively. Show that $OH\parallel BC$ if and only if $\tan{B}\tan{C}=3$.
Find the derivative of function $y=\sin{x}$.
Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$