`lim_(xto(pi)) (1-sin(x/2))/(cosx/2(cosx/4-sinx/4))`

Explanation

Given, `underset(xto(pi))"lim"(1-sin(x/2))/(cosx/2(cosx/4-sinx/4))`
`=underset(xto(pi))"lim"(cos^(2)x/4 + sin^(2)x/4-2.sinx/4.cosx/4)/(cosx/2.(cosx/4-sinx/4))` `[therefore sin^(2)theta+cos^(2)theta=1sin2theta=2sinthetacostheta]`
`=underset(xto(pi))"lim"(cosx/4-sinx/4)^(2)/((cos^(2)x/4-sin^(2)x/4)(cosx/4)-sin(x/4))` `[therefore cos^(2)2theta=cos^(2)theta-sin^(2)theta]`
`=underset(xto0)"lim"((cos(x/4)-sin(x/4))/((cosx/4+sinx/4)(cosx/4-sinx/4)))` `[therefore a^(2)-b^(2)=(a+b)(a-b)]`
=`underset(xto(pi))"lim"1/(cosx/4+sin x/4)=1/(1/sqrt(2)+1/sqrt(2))=sqrt(2)/2=1/sqrt(2)`