Solve each for all values of x

Question

√cos x = 2 cos x - 1

Answer 1

Let u = √(cosx).

Rewrite the equation as u = 2u² - 1.

Next, solve this equation for u (two answers).

 

For each answer for u, if it is positive, it's valid for that to be square root of some cosine.

 

So square that u, and then find inverse cosine of that value.  That is u = √(cosx) so

then u² = cos x  and x = arccos (u²).  Note; u squared will have to be ≤ 1.

 

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