Question 180507
You have to say this one in a way that makes sense
In words:
First, "arctan(x)" means "the angle whose tangent is x"
Therefore, "2*arctan(x)" means twice the angle whose tangent is x"
Now I can read the whole thing as:
"the sine of twice the angle whose tangent is x"
If the tangent is {{{x/1}}}, then the sine is {{{x/sqrt(x^2 + 1)}}}
and the cosine is {{{1/sqrt(x^2 + 1)}}}
The double angle formula for sine is:
{{{sin(2a) = 2*sin(a)*cos(a)}}}
{{{sin(2a) = 2*(x/sqrt(x^2 + 1))*(1/sqrt(x^2 + 1))}}}
{{{sin(2a) = (2x) / (x^2 + 1)}}} answer
Suppose the angle was 45 degrees. Then {{{x=1}}} and
{{{sin(2a) = (2x) / (x^2 + 1)}}}
{{{sin(90) = 2 / 2}}}
{{{sin(90) = 1}}} and this is true