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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