Solving a Trigonometric Equation: (cos x)^4 = cos (x^2)

[Xkaliber]

Hi all,

I was tutoring someone in Algebra III today and encountered a problem that threw me for a loop.

(cos x)^4 = cos (x^2)

Is there a way to solve this algebraically?

 

[Mentallic]

Hmm... looks so simple, yet it's so not

I wouldn't know of any algebraic methods to solve this.

y=cos⁴x is a nice, periodic function like any other standard trig function.

y=cos(x²) doesn't follow this pattern. As $$x \rightarrow \pm \infty$$ the period of fluctuations $$\rightarrow 0$$.

If there is an algebraic method to solve this - and I highly doubt it - then it can't be pretty.


Maybe starting with the simple answer x=0 and working from there (I suggest Newton's method ) would be a good start to approach this problem.