Show that: `tan(1/2sin^(-1)3/4)=(4\ sqrt(-7))/3`

Answer

N/a

Explanation

We have, `tan(1/2sin^(-1)3/4)=(4-sqrt(7))/(3)`
`:. LHS = tan[1/2sin^(-1)(3/4)]`
Let `1/2sin^(-1)'3/4=0rArrsin^(-1)'3/4 = 2theta`
`rArrsin2theta=3/4 rArr (2tan theta)/(1+tan^(2)theta) = 3/4`
`rArr 3+3tan^(2)theta=8tantheta`
`rArr 3tan^(2)theta - 8tan theta + 3 = 0`
Let ` tantheta = y`
`:. 3y^(2)-8tany+3=0`
`rArr y = (+8+-sqrt(64-4xx3xx3))/(2xx3)=(8+-sqrt(28))/(6)`
`= (2[4+-sqrt(7)])/(2.3)`
`rArr tantheta = (4+-sqrt(7))/(3)`
`rArr theta = tan^(-1)[(4+-sqrt(7))/(3)]`
{ but `(4+sqrt(7))/(3)gt1/2,pi/2`,max`[tan(1/2sin^(-1)'3/4)] = 1`}
`LHS = tan tan^(-1)((4-sqrt(7))/(3))=(4-sqrt(7))/(3) = RHS`