Question

How do you solve `tan(x+pi/6)=tan(x+pi/4)`?

Answer 1

There is no solution. The tangents of two arguments can be equal only if the arguments differ by a multiple of `\pi`.

Explanation:

Suppose `tan a=tan b`. Since

`sec^2 u=1/{cos^2 u}=1+tan^2 u`

we must have

`cos^2 a=cos^2 b`
`cos a=\pm cos b`

If `tan a=tan b` and `cos a=+cos b`, then `sin a=sin b`, and the equality of sines and cosines means the arguments `a` and `b` have to differ by an even number times `\pi`.

If `tan a=tan b` and `cos a=-cos b`, then `sin a=-sin b`, and the sines and cosines both being additive inverses means the arguments `a` and `b` have to differ by an odd number times `\pi`.

Either way the difference is some whole number times `\pi`.