Question

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

Answer 1

cos 3x

Explanation:

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

Answer 2

`+-2 x sqrt (1 - x^2 ) +- x sqrt(1 - 4 x^2 ), -1/2<=x<=1/2`

Explanation:

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

Let `a = arc cos ( 2 x )`.

Then, `cos a = 2 x in [-1, 1 ] and sin a = +- sqrt (1 - 2 x^2 )`

Note that `x in [-1/2. 1/2]`.

Prefix negative sign for principal value `a > pi/4` (when x > 0)..

Let `b = arc sin x`.

Then, `sin b = x and cos b = +- sqrt (1 - x^2 )`.

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

Now, the given expression is

`cos ( a + b )`

`= cos a cos b - sin a sin b`

`=+-2 x sqrt( 1 - x^2) +- x sqrt ( 1 - 4 x^2 )`