Question 942249
.
tan(x)=cot(4x+20°).  Find x.
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        The solution in the post by @lwsshar3 is a part of the truth, but not the whole truth.


        The solution by @lwsshar3 is correct if and only if both angles, x and 4x+20° are in QI.


        But if we look for all possible angles x between 0° and 180°  (180° is the period for both functions

        in the left and the right side of the equation, so it is NATURAL to investigate this interval), 

        then his solution is INCOMPLETE, because there are 5 (five) different angles 'x' satisfying the given equation 

        in interval [0°,180°).



        And this is the whole truth.  See my solution below.



<pre>
Equation

    tan(x) = cot(4x+20°)    (1)

implies

    x + (4x+20°) = 90° + k*180°, k = 0, +/-1, +/-2, . . . (2)


because both functions tan and cot are periodical with period {{{pi}}}.


So, we can rewrite equation (2) in the form

    5x = 70° + k*180°, k = 0, 1, 2, 3, 4.


We can limit ourselves with 5 values of k, because for other values of 'k', the angles 'x' will be geometrically the same,
but formally will be outside of the interval [0°,180°).


For these 5 values of 'k', we will have 5 different values of x in the interval [0°,180°)

    x = 14°,  14°+36° = 50°,  14°+72° = 86°,  14°+108° = 122°,  14°+144° = 158°.


<U>ANSWER</U>. Taking into account the periodicity of the left side and the right side of the given equation, 

it is natural to look for solutions 'x' in the interval [0°,180°).

In this interval, the given equation has 5 (five) different solutions 14°, 50°, 86°, 122° and 158°.
</pre>

Solved completely.



//////////// I N T E R E S T I N G ////////////



Driven by my curiosity, I posted today (May 21, 2026, about 2:00 PM) this problem to Google Overview Artificial Intelligence
to look how it treats the problem.


It treats it precisely as in the post by @lwsshar3 - so, its analysis is incomplete.


Does it embarrass me? - - - Both "Yes" and "No".


It simply demonstrates the level and the style of work of contemporary AI.


It can not think as a human can (because it is NOT ABLE to think as a human can), 
but simply re-writes from the storage of existing solutions, with no thinking.