To simplify this expression, we can use the angle addition formulas for sine and cosine.
sin(x) + sin(5x) + cos(x) + cos(5x) = 2sin(3x)cos(2x) + 2cos(3x)cos(2x)
= 2(cos(3x) + sin(3x))cos(2x)
= 2sqrt(2)cos(2x + pi/4)
Therefore, the expression simplifies to:
2sqrt(2)cos(2x + pi/4) = 0
To find the solutions for x, we can set cos(2x + pi/4) = 0:
2x + pi/4 = (2n + 1)pi/2, where n is an integer.
2x = (2n + 1)pi/2 - pi/4
x = (4n + 1)pi/4 - pi/8
Therefore, the general solution for x is:
x = (4n + 1)pi/4 - pi/8, where n is an integer.
To simplify this expression, we can use the angle addition formulas for sine and cosine.
sin(x) + sin(5x) + cos(x) + cos(5x) = 2sin(3x)cos(2x) + 2cos(3x)cos(2x)
= 2(cos(3x) + sin(3x))cos(2x)
= 2sqrt(2)cos(2x + pi/4)
Therefore, the expression simplifies to:
2sqrt(2)cos(2x + pi/4) = 0
To find the solutions for x, we can set cos(2x + pi/4) = 0:
2x + pi/4 = (2n + 1)pi/2, where n is an integer.
2x = (2n + 1)pi/2 - pi/4
x = (4n + 1)pi/4 - pi/8
Therefore, the general solution for x is:
x = (4n + 1)pi/4 - pi/8, where n is an integer.