Question

If $\sin^2 x + \sin x -1 = 0$, then value of the expression $[3\cos^6 x + 4\cos^4 x + 5\cos^2 x]$ is _____.

(where [.] represents greatest integer function)

Answer 1

5

Answer 2

Solve $\sin^2 x+\sin x-1=0$ to get $\sin x=\frac{\sqrt{5}-1}{2}$. Using $\cos^2x=1-\sin^2x$, we find $\cos^2 x=\frac{\sqrt{5}-1}{2}$. Let $u=\cos^2 x$. Calculate $u^2$ and $u^3$ and then evaluate $3u^3+4u^2+5u$ which simplifies to $\frac{7\sqrt{5}-5}{2}\approx5.326$. Hence the greatest integer is 5.