Question 1068467
Let X be the distance between the observer and the tower,
{{{tan(theta)=20/X}}}
{{{X=20/tan(theta)}}}
.
.
{{{tan(2*theta)=45/X}}}
{{{X=45/tan(2*theta)}}}
.
.

{{{20tan(theta)=45/tan(2*theta)}}}
{{{20tan(2*theta)=45tan(theta)}}}
Using an identity,
{{{20((2tan(theta))/(1-tan^2(theta)))=45tan(theta)}}}
Use a substitution for better readability,
{{{(40u)/(1-u^2)=45u}}}
{{{40u=45u-45u^3}}}
{{{45u^3-5u=0}}}
{{{5u(9u^2-1)=0}}}
Only the second solution is helpful,
{{{9u^2-1=0}}}
{{{9u^2=1}}}
{{{u^2=1/9}}}
{{{u=1/3}}}
{{{tan(theta)=1/3}}}
So then,
{{{X=20/tan(theta)}}}
{{{X=20/(1/3)}}}
{{{X=60}}}{{{m}}}