Practice Exercises for Trigonometry

I would like to know how this equality holds.
${\displaystyle {\tan }^{-1}\frac{(2n+1)-(2n-1)}{1+(2n+1)(2n-1)}={\tan }^{-1}\frac{1}{2n-1}-{\tan }^{-1}\frac{1}{2n+1}}$

I wanted to solve the following problem.
In ${\displaystyle \mathrm{△}ABC}$ we have
${\displaystyle {\sin }^{2}A+{\sin }^{2}B={\sin }^{2}C+\sin A\sin B\sin C}$
Compute ${\displaystyle \sin C}$
Since it's an equation for a triangle, I assumed that ${\displaystyle \pi =A+B+C}$ would be important to consider.
I've tried solving for ${\displaystyle \sin C}$ as a quadratic, rewriting ${\displaystyle \sin C=\sin (\pi -A-B)}$, but nothing seemed to work.
How does one approach this problem? Any help would be appreciated.

I want to solve
${\displaystyle \cos 3x-18\cos x+10=0}$

I want to prove this equality holds:
${\displaystyle \tan \left(\frac{\pi }{7}\right)\tan \left(\frac{2\pi }{7}\right)\tan \left(\frac{3\pi }{7}\right)=\sqrt{7}}$

I want to proof that
${\displaystyle 8{\cos }^4\left(\varphi \right)=\cos \left(4\varphi \right)+4\cos \left(2\varphi \right)+3}$

I want to calculate the antiderivative of this function .
${\displaystyle f\left(x\right)=\frac{\sqrt{\sin x-{\sin }^{3}x}}{\sqrt{1-{\sin }^{3}x}}dx}$
let ${\displaystyle \sqrt{\sin x-{\sin }^{3}x}=\sqrt{\sin x}\left|\cos x\right|}$ then I have :
${\displaystyle \int \frac{\sqrt{\sin x}\left|\cos x\right|}{\sqrt{1-{\sin }^{3}x}}dx}$ , then Assume ${\displaystyle \cos x>0}$ and if i take ${\displaystyle t=\sin x}$ and then using inverse function of sin it will be complicated , then is there any simple method to Evaluate it ?

I tried to solve a question which led to
${\displaystyle 4\sin \frac{\pi }{5}\sin \frac{3\pi }{5}=\sqrt{5}}$

I need to prove that
${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\sin 100n}$ diverges

I need to solve the following limit without using L'Hopital's rule:
${\displaystyle \underset{x\to 0}{lim}{(1+\sin \left(x\right))}^{\frac{1}{x}}}$
The thing is that I can not figure out what to do. One of my ideas was to apply this rule: ${\displaystyle a^x=e^{\ln \left(a^x\right)}=e^{x\cdot \ln \left(a\right)}}$ , getting this:
${\displaystyle \underset{x\to 0}{lim}{e}^{\frac{1}{x}\ln (1+\sin \left(x\right))}}$
I already know that the answer is e, so the exponent is definitely 1. However, I tried everything I could but have no idea how to solve ${\displaystyle \underset{x\to 0}{lim}\left(\frac{1}{x}\ln (1+\sin \left(x\right))\right)}$ which needs to be 1.
I would really appreciate your help, and if you find a totally different way to solve the limit without using L'Hospital's rule it will be good as well.

I need to solve for x in the polynomial
${\displaystyle 7{\sin }^2\left(x\right)-9\cos \left(2x\right)=0}$

I need to compute:
${\displaystyle \tan (\arctan \left(\frac{1}{2}\right)+\arctan \left(\frac{2}{9}\right)+\arctan \left(\frac{1}{8}\right)+\arctan \left(\frac{2}{25}\right)+\arctan \left(\frac{1}{18}\right)+\dots )}$

How to Simplify ${\displaystyle \frac{\sin \left(3x\right)+{\sin }^3\left(x\right)}{\cos \left(3x\right)-{\cos }^3\left(x\right)}}$
I tried to:
${\displaystyle \frac{\sin \left(2x\right)\cos \left(x\right)+\cos \left(2x\right)\sin \left(x\right)+{\sin }^3\left(x\right)}{\cos \left(2x\right)\cos \left(x\right)+\sin \left(2x\right)\sin \left(x\right)-{\cos }^3\left(x\right)}}$

How to prove this identity without using the actual values of ${\displaystyle {\tan 54}^{\circ }}$ and ${\displaystyle {\sin 24}^{\circ }}$
${\displaystyle {\tan 54}^{\circ }=\frac{{\sin 24}^{\circ }}{1-\sqrt{3}{\sin 24}^{\circ }}}$

How to prove the identity
${\displaystyle 3{\sin }^{4}x-2{\sin }^{6}x=1-3{\cos }^{4}x+2{\cos }^{6}x}$

How to prove the following equation by a quick method?
${\displaystyle {\sin }^{6}x+{\cos }^{6}x=\frac{1}{8}(3\cos 4x+5)}$

How to prove that ${\displaystyle \frac{1}{{\sin }^{2}x}+\frac{1}{{\sin }^{2}y}-\frac{2\cos (x-y)}{\left(\sin x\sin y\right)}=\frac{{\sin }^2(x-y)}{{\sin }^{2}x{\sin }^{2}y}}$

How to prove that ${\displaystyle \underset{x\to 0}{lim}\frac{x}{\sin x}=1}$, knowing that ${\displaystyle \underset{x\to 0}{lim}\frac{\sin x}{x}=1}$?

How to prove ${\displaystyle \cos (n!)\ne 1}$ without using ${\displaystyle \pi }$ is irrational

How to prove ${\displaystyle \sin 3\theta =3\sin \theta -4{\sin }^3\theta }$

How to prove ${\displaystyle {\cos 36}^{\circ }=\frac{1+\sqrt{5}}{4}}$