Calculate this sum :
you need to evaluate $\displaystyle cos(\frac{\pi}{9})$
To do this you need to consider $\displaystyle cos(a\pm b) = cos(a)\times cos(b)\mp sin(a)\times sin(b)$
where $\displaystyle cos(\frac{\pi}{9}) = cos(a\pm b)$
only after finding this will you be able to get an exact value
This technique will not work in this instance.
Edit: The off-topic discussion regarding the value of questions like this one can be found here: http://www.mathhelpforum.com/math-he...t-century.html. Any further discussion of this nature should be continued at that thread.
Or you can try
$\displaystyle
(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2) =
$
$\displaystyle
(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 +
(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 +
$
$\displaystyle (a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 +
(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2$
although I don't promise anything.
That thought had crossed my mind but it will then be necessary to solve the cubic equation $\displaystyle 4 t^3 - 3t - \frac{1}{2} = 0$ and I think that will be no mean feat (the method of Cardano could be used I suppose).
There will be a clever trick but I don't have time right.
Let us consider the equation
$\displaystyle cos3\theta=cos\frac{\pi}{3}$
there are infinite solutions of this equation
in the domain $\displaystyle [0,2\pi]$, the solutions are
$\displaystyle \theta=\frac{\pi}{9},\frac{5\pi}{9},\frac{7\pi}{9} ,\frac{11\pi}{9},\frac{13\pi}{9},\frac{17\pi}{9}.. ...............(a)$
The above two equations are identical. Every value of $\displaystyle \theta$ that satisfies equation 1 satisfies equation 2. If $\displaystyle \theta$ is a solution of equation 1, the $\displaystyle cos\theta$ is a root of the second equation.Equation1:
$\displaystyle cos3\theta=cos\frac{\pi}{3}$
Equation 2:
$\displaystyle 4cos^{3}x-3cosx=1/2$
For above solutions of $\displaystyle \theta$
$\displaystyle cos\theta=cos\frac{\pi}{9},cos\frac{5\pi}{9},cos\f rac{7\pi}{9},cos\frac{11\pi}{9},cos\frac{13\pi}{9} ,cos\frac{17\pi}{9}$
the last three of the above six are just the repetition of the first three. So the roots of Equation 2 can be
$\displaystyle cos\frac{\pi}{9}(=cos\frac{17\pi}{9}),cos\frac{5\p i}{9}(=cos\frac{13\pi}{9}),cos\frac{7\pi}{9}(=cos\ frac{11\pi}{9})$
Hence, we can say that $\displaystyle cos\frac{\pi}{9},cos\frac{5\pi}{9},cos\frac{7\pi}{ 9}$ are the three roots of equation 2.
Multipying equation 2 by $\displaystyle sec^{3}\theta$
we get
$\displaystyle sec^{3}\theta+6sec^{2}\theta-8=0$
roots are$\displaystyle sec\frac{\pi}{9},sec\frac{5\pi}{9},sec\frac{7\pi}{ 9}$
$\displaystyle sec\frac{\pi}{9}+sec\frac{5\pi}{9}+sec\frac{7\pi}{ 9}=-6$
$\displaystyle sec\frac{\pi}{9}.sec\frac{5\pi}{9}+sec\frac{7\pi}{ 9}.sec\frac{\pi}{9}+sec\frac{5\pi}{9}.sec\frac{7\p i}{9}=0$
the question of this thread is
$\displaystyle sec^{2}\frac{\pi}{9}+sec^{2}\frac{5\pi}{9}+sec^{2} \frac{7\pi}{9}+4$
$\displaystyle =(sec\frac{\pi}{9}+sec\frac{5\pi}{9}+sec\frac{7\pi }{9})^{2}-2(sec\frac{\pi}{9}.sec\frac{5\pi}{9}+sec\frac{7\pi }{9}.sec\frac{\pi}{9}+sec\frac{5\pi}{9}.sec\frac{7 \pi}{9})+4$
$\displaystyle =(-6)^{2}+2(0)+4$
$\displaystyle =40$