How do you simplify the expression $Sin(arctan(x)+arccos(x))$ ?

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Answer 1 · 2016-07-26T00:53:31

sin 2x

arctan x --> x
arccos x --> x
sin (arctan x + arccos x) = sin (x + x) = sin 2x

Answer 2 · 2016-07-26T04:08:42

. $=(1/sqrt(1+x^2))(sqrt(1-x^2)+-x^2)$ , x in [-1, 1]#.

The negative sign is used, when $x in [-1, 0]$ .

Let a = arc tan (x). The principal value of $a in [-pi/2, pi/2]$

Then x = tan a. sin a = $+-x/sqrt(1+x^2)$ and cos a = $1/sqrt(1+x^2)$ .

Let b = arc cos x. The principal value of $b in [0, pi]$

Then, x = cos b and sin b = $sqrt(1-x^2)$ . Also, $x in [-1, 1]$ .

The given expression =

$sin ( a + b )$

$= sin a cos b + cos a sin b$

$= (+-x/sqrt(1+x^2))(x)+(1/sqrt(1+x^2))sqrt(1-x^2)$

$=(1/sqrt(1+x^2))(sqrt(1-x^2)+-x^2)$ , x in [-1, 1]#.

The negative sign is used when $x in [-1, 0]$ .

How do you simplify the expression Sin(arctan(x)+arccos(x))Sin(arctan(x)+arccos(x))?