I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then $$ \pi\ge \frac{\pi}{x}\sin(x)\bigl(20+51\cos(x)-2\cos^2(x)+6\cos^3(x)\bigr)\bigl(21+18\cos(x)+36\cos^2(x)\bigr). $$ The usual method of showing that the left hand side equals $\pi$ for $x=0$ and then trying to prove that it decreases for $x>0$ by proving that the derivative is negative is difficult since to apply the derivative is quite complex.

I would appreciate any help.

I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then $$ \pi\ge \frac{\pi}{x}\sin(x)\bigl(20+51\cos(x)-2\cos^2(x)+6\cos^3(x)\bigr)/\bigl(21+18\cos(x)+36\cos^2(x)\bigr). $$ The usual method of showing that the left hand side equals $\pi$ for $x=0$ and then trying to prove that it decreases for $x>0$ by proving that the derivative is negative is difficult since to apply the derivative is quite complex.

I would appreciate any help.

I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then $$ \pi\ge \frac{\pi}{x}\sin(x)\big(20+51\cos(x)-2\cos^2(x)+6\cos^3(x)\big)\big(21+18\cos(x)+36\cos^2(x)\big) $$ The usual method of showing that the left hand side equals $\pi$ for $x=0$ and then trying to prove that it decreases for $x>0$ by proving that the derivative is negative is difficult since to apply the derivative is quite complex.

I would appreciate any help.

I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then $$ \pi\ge \frac{\pi}{x}\sin(x)\bigl(20+51\cos(x)-2\cos^2(x)+6\cos^3(x)\bigr)\bigl(21+18\cos(x)+36\cos^2(x)\bigr). $$ The usual method of showing that the left hand side equals $\pi$ for $x=0$ and then trying to prove that it decreases for $x>0$ by proving that the derivative is negative is difficult since to apply the derivative is quite complex.

I would appreciate any help.

I am preparing a paper on Huygens' approximations to pi and I have to prove the following inequality: if 0<= x<= pi/2 then

pi>= (pi/x)sin(x)(20+51cos(x)-2cos^2(x)+6cos^3(x))(21+18cos(x)+36cos^2(x))

The usual method of showing that the left hand side equals pi for x=0 and then trying to prove that it decreases for x>0 by proving that the derivative is negative is difficult since the derivative is quite complex.

I would appreciate any help.

I am preparing a paper on Huygens' approximations to $\pi$ and I have to prove the following inequality: if $0 \le x\le \pi/2$ then $$ \pi\ge \frac{\pi}{x}\sin(x)\big(20+51\cos(x)-2\cos^2(x)+6\cos^3(x)\big)\big(21+18\cos(x)+36\cos^2(x)\big) $$ The usual method of showing that the left hand side equals $\pi$ for $x=0$ and then trying to prove that it decreases for $x>0$ by proving that the derivative is negative is difficult since to apply the derivative is quite complex.

I would appreciate any help.