What is the general solution for tan(5x)+tan(3x)=0 ?

The general solution for tan(5x)+tan(3x)=0 is x=(pin)/4 ,x= pi/8+(pin)/4

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tan(5x)+tan(3x)=0

tan (5 x) + tan (3 x) = 0

x = \frac{πn}{4}, x = \frac{π}{8} + \frac{πn}{4}

Solution steps

tan (5 x) + tan (3 x) = 0

Express with sin, cos

\frac{cos ( 3 x ) sin ( 5 x ) + cos ( 5 x ) sin ( 3 x )}{cos ( 3 x ) cos ( 5 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (3 x) sin (5 x) + cos (5 x) sin (3 x) = 0

Rewrite using trig identities

sin (3 x + 5 x) = 0

General solutions for

sin (3 x + 5 x) = 0

3 x + 5 x = 0 + 2 πn, 3 x + 5 x = π + 2 πn

Solve

3 x + 5 x = 0 + 2 πn : x = \frac{πn}{4}

Solve

3 x + 5 x = π + 2 πn : x = \frac{π}{8} + \frac{πn}{4}

x = \frac{πn}{4}, x = \frac{π}{8} + \frac{πn}{4}

5 sin (2 x) = 5 cos (x)

2 cos^{2} (x) - 3 cos (x) - 5 = 0

cos (2 x) = - cos (x), 0 \leq x \leq 2 π

2 cos^{2} (w) + 3 cos (w) + 1 = 0

cos (3 x) + sin (3 x) = 1

What is the general solution for tan(5x)+tan(3x)=0 ?
The general solution for tan(5x)+tan(3x)=0 is x=(pin)/4 ,x= pi/8+(pin)/4