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tgx+tg4x=tg2x+tg3x equation

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Numerical solution:

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The solution

You have entered [src] 
tan(x) + tan(4*x) = tan(2*x) + tan(3*x)
tan⁡(x)+tan⁡(4x)=tan⁡(2x)+tan⁡(3x)\tan{\left(x \right)} + \tan{\left(4 x \right)} = \tan{\left(2 x \right)} + \tan{\left(3 x \right)}
Simplify
The graph
0-80-60-40-2020406080-100100-100000100000
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
  4*pi   3*pi   2*pi   pi   pi   2*pi   3*pi   4*pi     
- ---- - ---- - ---- - -- + -- + ---- + ---- + ---- + pi
   5      5      5     5    5     5      5      5
(((((((−4π5−3π5)−2π5)−π5)+π5)+2π5)+3π5)+4π5)+π\left(\left(\left(\left(\left(\left(\left(- \frac{4 \pi}{5} - \frac{3 \pi}{5}\right) - \frac{2 \pi}{5}\right) - \frac{\pi}{5}\right) + \frac{\pi}{5}\right) + \frac{2 \pi}{5}\right) + \frac{3 \pi}{5}\right) + \frac{4 \pi}{5}\right) + \pi 
=
pi
π\pi 
product
  -4*pi -3*pi -2*pi -pi  pi 2*pi 3*pi 4*pi   
0*-----*-----*-----*----*--*----*----*----*pi
    5     5     5    5   5   5    5    5
π4π53π52π5π5⋅−π5⋅−2π5⋅−3π5⋅0(−4π5)\pi \frac{4 \pi}{5} \frac{3 \pi}{5} \frac{2 \pi}{5} \frac{\pi}{5} \cdot - \frac{\pi}{5} \cdot - \frac{2 \pi}{5} \cdot - \frac{3 \pi}{5} \cdot 0 \left(- \frac{4 \pi}{5}\right) 
=
0
00 
0
Rapid solution [src] 
x1 = 0
x1=0x_{1} = 0 
     -4*pi
x2 = -----
       5
x2=−4π5x_{2} = - \frac{4 \pi}{5} 
     -3*pi
x3 = -----
       5
x3=−3π5x_{3} = - \frac{3 \pi}{5} 
     -2*pi
x4 = -----
       5
x4=−2π5x_{4} = - \frac{2 \pi}{5} 
     -pi 
x5 = ----
      5
x5=−π5x_{5} = - \frac{\pi}{5} 
     pi
x6 = --
     5
x6=π5x_{6} = \frac{\pi}{5} 
     2*pi
x7 = ----
      5
x7=2π5x_{7} = \frac{2 \pi}{5} 
     3*pi
x8 = ----
      5
x8=3π5x_{8} = \frac{3 \pi}{5} 
     4*pi
x9 = ----
      5
x9=4π5x_{9} = \frac{4 \pi}{5} 
x10 = pi
x10=πx_{10} = \pi 
x10 = pi
Numerical answer [src] 
x1 = -30.159289474462
x2 = 75.3982079073418
x3 = -79.7964534011807
x4 = 6.28317657745173
x5 = -74.1415866247191
x6 = 32.6725635973338
x7 = 36.4424747816416
x8 = 25.132766994109
x9 = 58.4336233567702
x10 = 80.4247719318987
x11 = 0.0
x12 = -11.9380520836412
x13 = 5.02654824574367
x14 = -81.6814273793448
x15 = 8.16814089933346
x16 = -3.76991118430775
x17 = -52.1504380495906
x18 = 89.2212313619501
x19 = -89.8495498926681
x20 = 40.8407145452695
x21 = 72.2566292948604
x22 = -96.1327351998477
x23 = 91.734505484822
x24 = -45.867252742411
x25 = -19.4778744522567
x26 = 64.0884901332318
x27 = -13.8230076757951
x28 = -37.6991253259725
x29 = -64.0884901332318
x30 = 87.9646065891021
x31 = 98.0176907920015
x32 = -99.9026463841554
x33 = -18.2212373908208
x34 = -38.9557489045134
x35 = 30.159289474462
x36 = -43.9823032737618
x37 = 65.973454934889
x38 = 64.7168086639497
x39 = 62.2035345410779
x40 = 84.1946831162065
x41 = -87.9646063065009
x42 = -9.42481832724324
x43 = 50.2654783980164
x44 = 42.0973415581032
x45 = 86.0796387083603
x46 = -23.8761041672824
x47 = 10.0530964914873
x48 = -69.7433569096934
x49 = 28.2743274943129
x50 = -67.8584013175395
x51 = -57.8053048260522
x52 = -59.6902763293664
x53 = 43.9823032903129
x54 = -83.5663645854885
x55 = 54.6637121724624
x56 = -33.9292006587698
x57 = 21.9911516466404
x58 = -25.7610597594363
x59 = -61.5752160103599
x60 = -1.88495559215388
x61 = -42.0973415581032
x62 = -7.5398223686155
x63 = 18.2212373908208
x64 = 51.5221195188726
x65 = -47.7522083345649
x66 = 94.2477801895162
x67 = 14.451326206513
x68 = -21.9911516456241
x69 = -65.9734548486986
x70 = -35.8141562509236
x71 = -91.734505484822
x72 = -15.707974359936
x73 = -98.0176907920015
x74 = 40.2123859659494
x75 = 20.1061929829747
x75 = 20.1061929829747
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