Find the general solution for the equation 4tanx+cotx=5?????? and the steps please? THANYOO!

Answer 1

For these I generally like to change everything to sines and cosines

4tanx + cotx = 5
4sinx/cosx + cosx/sinx = 5
(4sin^2x + cos^2x)/sinxcosx = 5
4sin^2x + cos^2x = 5sinxcosx
4sin^2x - 5sinxcosx + cos^2x = 0
(4sinx - cosx)(sinx - cosx) = 0

4sinx = cosx
sinx/cosx = 1/4
tanx = 1/4
x = 14.0° or 194.0°

or

sinx = cosx
sinx/cosx = 1
tanx=1
x = 45° or 225°

the period of tanØ is 180°

general solutions:
45° + 180k°
14° + 180k°

notice that 45+180 = 225 , the other answer
and 14+180 = 194 , also the other answer, so we don't have to generalize those, they get hit anyway.