Question #381cb

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Answer 1 · 2018-01-17T06:06:22

# x=125/117#.

Prequisites :

#(1):tan^-1x+tan^-1y=tan^_1((x+y)/(1-xy)); x,y gt 0, xy lt1#.

#(2):cos^-1x=tan^-1(sqrt(1-x^2)/x), 0 lt x lt 1#.

#(3):tan^-1x=sin^-1(x/sqrt(1+x^2)), 0ltxlt1#.

Therefore, #2tan^-1(1/7)+cos^-1(3/5)#,

#={tan^-1(1/7)+tan^-1(1/7)}+cos^-1(3/5)#,

#=tan^-1{(1/7+1/7)/(1-1/7*1/7)}+tan^-1{sqrt(1-(3/5)^2)/(3/5)}#,

#=tan^-1(2/7*49/48)+tan^-1(4/5*5/3)#,

#=tan^-1(7/24)+tan^-1(4/3)#,

#=tan^-1{(7/24+4/3)/(1-7/24*4/3)}#,

#=tan^-1(117/44)#,

#=sin^-1{(117/44)/sqrt(1+(117/44)^2)}#,

#=sin^-1(117/44*44/125)#,

#=sin^-1(117/125)#.

#:. sin^-1(1/x)=sin^-1(117/125)#.

#rArr x=125/117#.

Since, the eqn. involves trigo. inverse funs., the root needs

verification, which I leave to the Questioner.