Q.344. Trigonometric equations

Question 344.
Find general solution of sin(5x) = tan(x).

Answer 344.
sin5x
= sin5x - sinx + sinx
= 2cos3x sin2x + sinx
= 2(4cos^3 x - 3cosx) * 2sinx cosx + sinx
= sinx (16cos^4 x - 12cos^2 x + 1)

Hence,
sin5x = tanx
=> sin5x - sinx/cosx = 0
=> sin5x cosx - sinx = 0
=> sinx (16cos^5 x - 12cos^3 x + cosx - 1) = 0
=> sinx = 0
=> x = kπ
or 16cos^5 x - 12cos^3 x + cosx - 1 = 0
As this is not easily factorizable, using Wolfram Alpha (link as under)
=> x = 2kπ ± 0.509694
=> x = {kπ, 2kπ ± 0.509694, k ∈ Z} is the general solution.
Source(s):
http://www.wolframalpha.com/input/?i=16cos%5E5+x+-+12cos%5E3+x+%2B+cosx+-+1+%3D+0

Link to YA!