We have ,

`tan(A+B-C)=1=tan45^(@)`

`rArrA+B-C=45^(@)` . . . (1)

Also `sec(B+C-A)=2sec60^(@)`

`rArr B+C-A=60^(@)` . . .(2)

Adding equations (1) and (2) we get

`2B=105^(@) rArr B=52.5^(@)` . . .(3)

Subtracting equation (1) from equation (2) , we get

`2C-2A=15^(@)rArrC-A=7.5^(@)` . . . (4)

we know that A+B+C `=180^(@)`

`rArrA+C=180^(@)-52.5^(@)` [from(3)]

`rArrA+C=127.5^(@)` . . .(5)

`rArr-A+C=7.5^(@)` . . . (4)

Adding equations (4) and (5) , we get

`2C=135^(@)rArrC=67.5^(@)`

`:.A=127.5^(@)-67.5^(@)=60^(@)`

`:. cos (4B-3A)=cos(4xx52.5^(@)-3xx60^(@))=cos(210^(@)-180^(@))=cos30^(@)`

`=(sqrt(3))/(2)`