find the exact value of cos arctan 34 arcsin 13 got triang

Solution

cos [ arctan (-3/4) + arcsin (1/3)]

Let x = arctan (-3/4) ; y = arcsin (1/3)

cos(x+y) = cosx cosy -sinxsiny

x = arctan (-3/4)

tanx = -3/4 ----> sinx = -3/5

x= arcsin(-3/5) ;

cosx = 4/5 ---> x = arccos(4/5)

y = arcsin (1/3)

siny = 1/3 ; cosy = 2sqrt2/3

cos [ arctan (-3/4) + arcsin (1/3)] = cosx cosy -sinxsiny

= (4/5)*2sqrt2/3 - (-3/5)(1/3)

= { 8sqrt(2) + 3 }/15