Question 476243
arccos(4/5) equals the angle which satisfies the equation: {{{cos(phi) = 4/5}}}
arcsin(24/25) equals the angle which satisfies  the equation: {{{sin(theta) = 24/25}}}
So we are proving that {{{2phi = theta}}}
Lets use a trig property for double angles.
{{{cos(2phi) = 2*cos^2(phi) - 1}}}
Substitute {{{cos(phi) = 4/5}}}
{{{cos(2phi) = 2*(4/5)^2 - 1}}}
{{{cos(2phi) = 2*(16/25) - 1}}}
{{{cos(2phi) = 32/25 - 1}}}
{{{cos(2phi) = 32/25 - 25/25 = 7/25}}}
Now assume {{{2phi = theta}}} is true and check if it works.
{{{cos(2phi) = cos(theta) = 7/25}}}
Now use a trig identity:
{{{sin^2(theta) + cos^2(theta) = 1}}}
{{{(24/25)^2 + (7/25)^2 = 1}}}
{{{(24^2 + 7^2)/25^2 = 1}}}
{{{625/625 = 1}}}
It works
Therefore {{{2phi}}} is equivalent to {{{theta}}}.