If `tanalpha=1/(1+2^(-x)) and tanbeta=1/(1+2^(x+1)),` then write the value of `alpha+beta` lying in the interval `(0,pi//2)` .

Explanation

We know that `tan(alpha+beta)=(tan(alpha)+tan(beta))/(1-tan(alpha).tan(beta))`

=`(1/(1+2^(-x))+1/(1+2^(x+1)))/(1-1/(1+2^(-x)).1/(1+2^(x+1)))`

=`(1+2^(x+1)+1+2^(-x))/(1+2^(x+1)+2^(-x)+2^(-x+x+1)-1)`

=`(2+2^(x+1)+2^(-x))/(2+2^(x+1)+2^(-x))`

=`1`

Therefore `alpha+beta=tan^(-1)(1)=pi/4`