How do you simplify the expression $cos(arctan 2x - arcsin x)$ ?

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Answer 1 · 2016-08-25T05:09:17

$(sqrt(1-x^2)+2x^2)/sqrt(1+4x^2)$

Let $a = arc tan 2x in Q1 or Q4$ , wherein cosine is positive.

Then, $tan a = 2x, sin a = (2x)/sqrt(1+4x^2) and cos a = 1/sqrt(1+4x^2)$

Let $b = arc sin x in Q1 or Q4$ , wherein cosine is positive.

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

Now, the given expression is

$cos(a-b)$

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

$=(1/sqrt(1+4x^2))(sqrt(1-x^2))+((2x)/sqrt(1+4x^2))(x)$

$=(sqrt(1-x^2)+2x^2)/sqrt(1+4x^2)$ .

See how it works.

Let $x = 1/2$ . Then $a = arc sin (1/2) = pi/6 and b = arc tan 1 = pi/4$

$cos (a-b) = cos (pi/4-pi/6)=cos (pi/12) = (sqrt3 + 1 )/(2sqrt2)$

When $x = 1/2$ ,

$(2x^2+sqrt(1-x^2))/sqrt(1+4x^2)$

#=(1/2+sqrt3/2)/sqrt(1+1)@

$= (sqrt3 + 1 )/(2sqrt2)$

How do you simplify the expression cos(arctan 2x - arcsin x)cos(arctan 2x - arcsin x)?