What is the general solution for sin(5x)+sin(x)=0 ?

The general solution for sin(5x)+sin(x)=0 is x= pi/4+pin,x=(3pi)/4+pin,x=(2pin)/3 ,x= pi/3+(2pin)/3

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sin(5x)+sin(x)=0

sin (5 x) + sin (x) = 0

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

Solution steps

sin (5 x) + sin (x) = 0

Rewrite using trig identities

2 cos (2 x) sin (3 x) = 0

Solving each part separately

cos (2 x) = 0 or sin (3 x) = 0

cos (2 x) = 0 : x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

sin (3 x) = 0 : x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

cot^{2} (x) + csc^{2} (x) = 3

2 tan (x) = 0

5 cos (2 θ) = 15 cos (θ) - 10

cos^{2} (x) = 1 + sin^{2} (x)

2 cos^{2} (x) + cos (x) = 0, 0 \leq x \leq 2 π

What is the general solution for sin(5x)+sin(x)=0 ?
The general solution for sin(5x)+sin(x)=0 is x= pi/4+pin,x=(3pi)/4+pin,x=(2pin)/3 ,x= pi/3+(2pin)/3