Question

Express the equation `tan(theta+45)-2tan(theta-45)=4` as a quadratic equation in `tan theta`. Hence solve the equation. Can someone please solve this?

Answer 1

`theta=nxx180^@+28.6742^@`

or `theta=nxx180^@-14.6386^@`

where `n` is an integer.

Explanation:

`tan(theta+45^@)-2tan(theta-45^@)=4` can be written as

`(tantheta+tan45^@)/(1-tanthetatan45^@)-2*(tantheta-tan45^@)/(1+tanthetatan45^@)=4`

or `(tantheta+1)/(1-tantheta)-2*(tantheta-1)/(1+tantheta)=4`

or `(tantheta+1)^2+2(1-tantheta)^2=4-4tan^2theta`

or `tan^2theta+2tantheta+1+2+2tan^2theta-4tantheta=4-4tan^2theta`

or `7tan^2theta-2tantheta-1=0`

and using quadratic formula `tantheta=(2+-sqrt(4+28))/14`

= `(1+-2sqrt2)/7` i.e. `0.5469` or `-0.2612`

and `theta=tan28.6742^@` or `-14.6386^@`

and general solution is `theta=nxx180^@+28.6742^@` or `theta=nxx180^@-14.6386^@`, where `n` is an integer.