Evaluate the following limit: `lim_(x->pi//2)(sqrt(2)-sqrt(1+sin x))/(cos^2x)`

Explanation

`lim_(x->pi//2)(sqrt(2)-sqrt(1+sin x))/(cos^2x)`

`=lim_(x->pi//2)(sqrt(2)-sqrt(1+sin x))/(cos^2x) xx (sqrt2+sqrt(1+sinx))/(sqrt2+sqrt(1+sinx))`

`=lim_(x->pi//2) (1-sin x)/((1+sin x)(1-sin x)(sqrt2 +sqrt(1+sin x)))`

`=1/(2(2sqrt2))`

`=1/(4sqrt2)`