Find exact value Thanks Find exact value sin cos1 1213 tan

Find exact value. Thanks =)

Find exact value. sin (cos^-1 12/13 + tan^-1 4/3)

given problem is sin ( arc cos(12/13) + arc tan(4/3) )

now assume arc cos(12/13) = A and arc tan(4/3) = B

so now 12/13 = cosA and 4/3 = tan(B)

now the given problem becomes sin(A+B) [ since we assume arc cos(12/13) = A and arc tan(4/3) = B ]

we know the formula of sim(A+B) = sinA. cosB + cosA.sinB -------->equation1

we know that cosA = 12/13 then sinA = sqrt[1 -(12/13)^2]

sinA = sqrt[1 -144/169]

sinA= sqrt(25/169)

sinA= 5/13

we know tanB = 4/3

so now sinB = 4/5 and cosB=3/5

now plug sinA= 5/13, cosA= 12/13 and sinB = 4/5, cosB=3/5 in equation1

sin(A+B) = sinA. cosB + cosA.sinB

= 5/13 . 3/5 + 12/13. 4/5

= 15/65 + 48/65

= 63/65

so sin ( arc cos(12/13) + arc tan(4/3) ) = 63/65