Give the exact real number value or an algebraic expression

Give the exact real number value or an algebraic expression for the following. sin (2 tan^-1 12/5) sin (sec^-1 u/2)

Solution

a)sin(2tan^-112/5)

sin2x =2sinx cosx

=2sin(tan^-112/5)cos(tan^-112/5)

hypotenuse =sqrt[oppositeside²+adjacent side²]=sqrt[12²+5²]=13

=2sin(sin^-112/13)cos(cos^-15/13)

=2(12/13)(5/13)

=120/169

b)sin(sec^-1(u/2))

sin(cos^-1(2/u))

oppositeside =sqrt[hypotenuse² -adjacent side²]=sqrt[u²-2²]=sqrt[u²-4]

sin(sin^-1(sqrt[u²-4]/u))

=(sqrt[u²-4])/u