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When working with spherical coordinates I have come across
${\displaystyle {\cos }^{-1}(\cos \left(a\right)\times \cos \left(b\right))}$
and was wondering if there was some kind of identity for this that I could use instead of computing the inside of the acrccos function first.

After much of solving, I arrive at the expression
${\displaystyle \frac{1}{2}[13\cos A-3\sqrt{3}\sin A+6]}$

What is the period for the following:
${\displaystyle y=10\sin \left(\frac{2\pi }{365}(x-50)\right)}$
Is the period
${\displaystyle \frac{2\pi }{\frac{2\pi }{365}}}$
which would be ${\displaystyle 365}$ ?

Why are these ${\displaystyle \sum \cos }$ and ${\displaystyle \csc }$ equivalent?
Mathematica 'simplifies' this formula
${\displaystyle \sum _{k=1}^R\cos \frac{2k\pi x}{R}}$
to this
${\displaystyle \frac{1}{2}(\csc \frac{\pi x}{R}\sin \frac{(2R+1)\pi x}{R}-1)}$

Kimberly took her 6 nieces and nephews to a hockey game. She wants to buy them snacks. How much can she spend on snacks for each child if Kimberly wants to spend less than $33 in total?

What is the value of
${\displaystyle {\csc }^2\left(\frac{\pi }{7}\right)+{\csc }^2\left(\frac{2\pi }{7}\right)+{\csc }^2\left(\frac{4\pi }{7}\right)}$

What is the value of ${\displaystyle {\int }_{\{-\mathrm{\infty }\}}^{\mathrm{\infty }}{\cos }^{2}xdx}$?

What is the value of ${\displaystyle {\sin }^2\left(\frac{\pi }{10}\right){\sin }^2\left(\frac{3\pi }{10}\right)}$?

When does ${\displaystyle {\arcsin }^{2}x+{\arccos }^{2}x=1}$ for x real or comlex?

Why is ${\displaystyle 2^{\frac{n}{2}}{\left(\frac{5}{3}\right)}^n(\cos \left(n\frac{\pi }{4}\right)+\sin \left(n\frac{\pi }{4}\right))}$ an alternate form of the complex number ${\displaystyle {\left(\frac{5}{3}\right)}^n{(1+i)}^n}$ ?

Why is ${\displaystyle {\int }_{-\pi }^{\pi }(2-\left|x\right|)\cos \left(x\right)=4}$ ?

What is the value of ${\displaystyle Arg\left(\frac{z}{\bar{z}}\right)}$?
As part of an exercise, I was asked to plot this condition on the complex plane:
${\displaystyle -\frac{\pi }{2}\le \text{ Arg }\left(\frac{z}{\bar{z}}\right)\le \frac{\pi }{2}}$

What is the value of ${\displaystyle {\sin 47}^{\circ }+{\sin 61}^{\circ }-{\sin 25}^{\circ }-{\sin 11}^{\circ }}$

What is x, if ${\displaystyle {\cot }^{-1}(3x+\frac{2}{x})+{\cot }^{-1}(6x+\frac{2}{x})+{\cot }^{-1}(10x+\frac{2}{x})+\cdots =1}$

Why
${\displaystyle {\cos }^{3}x-2\cos \left(x\right){\sin }^2\left(x\right)=\frac{1}{4}(\cos \left(x\right)+3\cos \left(3x\right))}$

Why can I cancel this ${\displaystyle \sum _{i}2{\cos }^3\left({\alpha }_i\right){\sin \alpha }_i}$ term for pairs 180degrees apart?
I have the equation,
${\displaystyle K_{11}\sum _i{\cos }^4{\alpha }_i+K_{12}\sum _{i}2{\cos }^3{\alpha }_i{\sin \alpha }_i+K_{22}\sum _i{\sin }^2{\alpha }_i{\cos }^2{\alpha }_i=\sum _i{\rho }_i{\cos }^2{\alpha }_i}$
In the paper it states, "For equal intervals in 180 degree segments the sums of odd powers are zero; therefore,"
${\displaystyle K_{11}\sum _i{\cos }^4{\alpha }_i+K_{22}\sum _i{\sin }^2{\alpha }_i{\cos }^2{\alpha }_i=\sum _i{\rho }_i{\cos }^2{\alpha }_i}$
The way I interpret what they mean is that, if I sum the values of ${\displaystyle {\cos }^3{\alpha }_i}$ for 0 degree and 180 degree I get a value of 0. Similarly, if I sum the values of ${\displaystyle {\cos }^3{\alpha }_i}$ for 1 degrees and 181 degree I get a value of 0. And so on and so forth all the way up to 179 degree and 359 degree, the sum of the values of ${\displaystyle {\cos }^3{\alpha }_i}$ will be 0. Therefore, we can cancel out the ${\displaystyle K_{12}\sum _{i}2{\cos }^3{\alpha }_i{\sin \alpha }_i}$ term of Eqn(1). However, if I were to make the same sums with the term ${\displaystyle \sum _{i}2{\cos }^3{\alpha }_i{\sin \alpha }_i}$ I see that I do not get values of zero.

Why is ${\displaystyle {\cos }^2\left(2\frac{\pi }{5}\right)+{\cos }^2\left(4\frac{\pi }{5}\right)=\frac{3}{4}}$ ?

Why is ${\displaystyle x^3-5x}$ injective on the rationals?

Why is ${\displaystyle \frac{0.5}{{\cos }^2\left(30\right)}=\frac{\tan \left(30\right)}{\cos \left(30\right)}}$, but ${\displaystyle \frac{0.5}{{\cos }^2\left(13\right)}\ne \frac{\tan \left(13\right)}{\cos \left(13\right)}}$?

Consider the set s =(x:x is a multiple of 10,x<30),described the set in words