Trigonometry Problems - Basics, Advanced & Examples

How to go from ${\displaystyle 2\cos 2x+1\to \frac{\sin 3x}{\sin x}}$

How do you find the limit of ${\displaystyle \underset{x\to 0}{lim}\frac{\tan x-\sin x}{x^3}}$?
My attempt:
${\displaystyle \underset{x\to 0}{lim}\frac{\tan x-\sin x}{x^3}}$
${\displaystyle \underset{x\to 0}{lim}\frac{1}{x^2}(\frac{\tan x}{x}-\frac{\sin x}{x})}$
${\displaystyle \underset{x\to 0}{lim}\frac{1}{x^2}(1-1)}$
${\displaystyle \underset{x\to 0}{lim}\frac{1}{x^2}\times 0}$
0
However, according to wolframalpha and my book, I'm wrong.

Eliminating ${\displaystyle \theta }$ from ${\displaystyle x=\cos \theta (2-\cos 2\theta )}$ and ${\displaystyle y=\sin \theta (2-\sin 2\theta )}$

I am having trouble finding the correct answer to this question:
Find the angles between ${\displaystyle -{360}^{\circ }\text{and}{180}^{\circ }}$ such that ${\displaystyle \sqrt{2}\sin ({90}^{\circ }-x)+1=0}$

How do we show ${\displaystyle 1-\cos x\ge \frac{x^2}{3}}$ for ${\displaystyle \left|x\right|\le 1}$? My first idea was to write
${\displaystyle 1-\cos x=\frac{1}{2}{|e^{ix}-1|}^2}$
which is true for all ${\displaystyle x\in \mathbb{R}}$, but I don't have a suitable lower bound for the right-hand side at hand.

Eliminate ${\displaystyle \theta \text{and}\varphi }$ between the following equations:
${\displaystyle \{\begin{array}{c}\sin \theta +\sin \varphi =x\\ \cos \theta +\cos \varphi =y\\ \tan \frac{\theta }{2}\tan \frac{\varphi }{2}=z\end{array}}$
What Ive

Determining the period of ${\displaystyle \frac{\sin 2x}{\cos 3x}}$
I would like to compute the period of this function which is a fraction of two trigonometric functions.
${\displaystyle \frac{\sin 2x}{\cos 3x}}$
Is there a theorem for this? what trick to use to easily find the period? I started by reducing the fraction but Im

Finding solution of a Trigonometric equation
${\displaystyle \tan A+\tan 2A+\tan 3A=0}$
I tried converting these all in sin and cos and I got the answer but the answer didn't match

Solve this trigonometric equation ${\displaystyle \sin 2x-\sqrt{3}\cos 2x=2}$
I tried dividing both sides with ${\displaystyle \cos 2x}$ but then I win ${\displaystyle \frac{2}{\cos 2x}}$

How do you simplify ${\displaystyle \sin \left({\tan }^{-1}\left(x\right)\right)}$?

What is ${\displaystyle \sin \left(x\right)+\cos \left(x\right)}$ in terms of sine?

Prove that ${\displaystyle \sin x+\cos x=\sqrt{2}\sin (x+\frac{\pi }{4})}$
${\displaystyle \sqrt{2}(x+\frac{\pi }{4})=\sqrt{2}(\sin x\cos \frac{\pi }{4}+\cos x\sin \frac{\pi }{4})=\sin x+\cos x}$
Could you solve it from opposite?
${\displaystyle \sin x+\cos x=\sqrt{2}\sin (x+\frac{\pi }{4})}$

Evaluating ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{(1+\frac{\sin n}{5n+1})}^{2n+3}}$ (a ${\displaystyle 1^{\mathrm{\infty }}}$ indeterminate form)

Why is ${\displaystyle \tan \theta \approx \frac{1}{\frac{\pi }{2}-\theta }}$ for ${\displaystyle \theta }$ close to ${\displaystyle \frac{\pi }{2}}$?
I wanted to see what the behaviour of the steep part of the ${\displaystyle \tan }$ curve was like, i.e. the behaviour of ${\displaystyle \tan \left(x\right)asx\to {\left(\frac{\pi }{2}\right)}^{-}}$. So by thinking about a shift of the graph of ${\displaystyle \tan \left(x\right)by\frac{\pi }{2}}$ to the left, I put some small (positive and negative) values of ${\displaystyle \theta }$ into my calculator for the function ${\displaystyle \tan (\theta +\frac{\pi }{2})}$
${\displaystyle \tan \theta \approx \frac{1}{\frac{\pi }{2}-\theta }}$ for ${\displaystyle \theta }$ close to ${\displaystyle \frac{\pi }{2}}$?
or, in more colloquial terms,
The steep part of ${\displaystyle \tan x}$ is just like the steep part of ${\displaystyle \frac{1}{x}}$
But why is this the case? I couldn't deduce it easily using the Maclaurin expansion of ${\displaystyle \tan \left(x\right)}$. Is there a more intuitive explanation? I couldn't think of any explanations analogous to those explaining small angle approximations.

How do you calculate the slope, x intercept, and y intercept of the following equation: ${\displaystyle f\left(x\right)=-6x-3}$?

What does ${\displaystyle \cos x\sin x}$ equal?

How do i solve cos (-60(degrees)) to exact values?

Find the value of ${\displaystyle \cos 240°}$.

What is the domain and range of ${\displaystyle y=\arcsin x}$

If ${\displaystyle \theta }$ is an acute angle and ${\displaystyle \sin \theta =\frac{1}{2}}$, then what is ${\displaystyle \cos \theta }$?