The value of `lim_(xrarrpi//4) (tan^(3)x-tanx)/(cos(x+(pi)/(4)))` is

Options

A. 8
B. 4
C. `-8`
D. `-2`

Answer

C

Explanation

We have, `underset(xrarrpi//4)(lim)(tan^(3)x-tanx)/(4cos(x+pi//4))`
`=underset(xrarrpi//4)(lim)(tanx(tanx-1)(tanx+1))/(cos(x +pi//4))`
`=underset(xrarrpi//4)(lim)(tanx(sinx-cosx)(tanx+1))/(cosx cos(x+pi//4))`
`=-underset(xrarrpi//4)(lim)(tanx(cosx-sinx)(tanx+1))/(cosx cos(x+pi//4))`
`=-sqrt2underset(xrarrpi//2)(lim)(tanx((1)/(sqrt2)cosx-(1)/(sqrt2)sinx)(tanx+1))/(cosx cos(x+pi//4))`
`=-sqrt2underset(xrarrpi//4)(lim)(tanx(tanx+1))/(cosx)`
`=-sqrt2xx2xxsqrt2=-8`