why is $$\frac{d}{dx}\cos(x)=-\sin(x)$$ I am studying for a differential equation test and I seem to always forget \this, and i am just wondering if there is some intuition i’m missing, or is it just one of those things to memorize? and i know this is not very differential equation related, just one of those things […]

$9$. Consider the parametric curve $K\subset R^3$ given by $$x = (2 + \cos(2s)) \cos(3s)$$ $$y = (2 + \cos(2s)) \sin(3s)$$ $$z = \sin(2s)$$ a) Express the equations of K as polynomial equations in $x,\ y,\ z,\ a = \cos(s),\ b = \sin(s)$. Hint: Trig identities. b) By computing a Groebner basis for the ideal […]

Prove that for a natural number $n$, $$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$

Prove the inequality: $$\left(1+\dfrac{1}{\sin a}\right)\left(1+\dfrac{1}{\cos a}\right)\ge 3+2\sqrt{2}; \text{ for } a\in\left]0,\frac{\pi}{2}\right[$$

Could anyone just evaluate this expression? I would be very grateful. $$\cot^2\left(\frac{1\pi}{13}\right)+\cot^2\left(\frac{2\pi}{13}\right)+\cot^2\left(\frac{3\pi}{13}\right)+\cot^2\left(\frac{4\pi}{13}\right)+\cot^2\left(\frac{5\pi}{13}\right)+\cot^2\left(\frac{6\pi}{13}\right)$$

The figure below is a trapezoid, what is the length of the red line? Thank you very much in advance!

Find the value of $$\sum_{n=1}^{50}\arctan\left(\frac{2n}{n^4-n^2+1}\right)$$ $$\frac{2n}{n^4-n^2+1}=\frac{2n}{1-n^2(1-n^2)}$$ I am not able to split it into sum or difference of two $\arctan$s.Please help me.

As the title says.. it says to use the mean value theorem but I don’t see how that’s applicable. Thank you

Suppose $x \in (0,\frac{\pi}{2})$ and we want to prove $$x<\frac{\sin x +\tan x}{2}$$I tried to prove it by taking $f(x)=\sin x+ \tan x -2x$ and show $f(x) >0 ,when\space x \in (0,\frac{\pi}{2})$ take f’$$f’=\cos x +1+\tan ^2 x-2\\=\tan^2 x-(1-\cos x)\\=\tan ^2 x-2sin^2(\frac x2)$$ I get stuck here ,because the last line need to be proved […]

There’s a math clock with formulas for each of $1,\ldots,12$, most of which are easy. Number 11, however, intrigues me: $$10^{12} + 10^7 – 45\sum_{k=1}^{999}\csc^4\frac{k\pi}{1000}.$$ Wolfram Alpha agrees the answer is (around) 11. How does one prove this? How does one come up with this?

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