How do you simplify cos(arccos(2x)+arcsin(x))?

2 Answers
Jul 24, 2016

cos 3x

Explanation:

arccos 2x --> 2x
arcsin x --> sin x
cos (arccos 2x + arcsin x) = cos (2x + x) = cos 3x

Jul 24, 2016

±2x1x2±x14x2,12x12

Explanation:

arc cos 2x is the angle whose cosine is 2x.

Let a=arccos(2x).

Then, cosa=2x[1,1]andsina=±12x2

Note that x[12.12].

Prefix negative sign for principal value a>π4 (when x > 0)..

Let b=arcsinx.

Then, sinb=xandcosb=±1x2.

Prefix negative sign for principal value b<0 (when x < 0).

Now, the given expression is

cos(a+b)

=cosacosbsinasinb

=±2x1x2±x14x2