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

1 Answer
Oscar L.
Jul 22, 2016

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.

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