The value of `lim_(x->1/(sqrt(2))) ((x-"cos" (sin^(-1)x))/ (1-tan(sin^(-1)x)))` is

(a) `-1/(sqrt(2))`

(b) `1/(sqrt(2))`

(c) `sqrt(2)`

(d) `-sqrt(2)`

Options

A. `-(1)/(sqrt(2))`
B. -1
C. non-existent
D. none of these

Answer

A

Explanation

`Let sin^(-1)x=theta.Then,x=sintheta.`
Now, `xto(1)/(sqrt(2))"or " thetato(pi)/(4)." Therefore,"`
`underset(xto(1)/(sqrt(2)))lim(x-cos(sin^(-1)x))/(1-tan(sin^(-1)x))=underset(thetato(pi)/(4))lim(sintheta-costheta)/(1-tantheta)`
`=underset(thetato(pi)/(4))lim((sintheta-costheta))/((costheta-sintheta))costheta`
`=underset(thetato(pi)/(4))lim-costheta=-(1)/(sqrt(2))`