What is the general solution for tan(2x)-1/(tan(x))=0 ?

The general solution for tan(2x)-1/(tan(x))=0 is x=0.52359…+pin,x=-0.52359…+pin

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tan(2x)-1/(tan(x))=0

tan (2 x) - \frac{1}{tan ( x )} = 0

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

Solution steps

tan (2 x) - \frac{1}{tan ( x )} = 0

Simplify

tan (2 x) - \frac{1}{tan ( x )} : \frac{tan ( 2 x ) tan ( x ) - 1}{tan ( x )}

\frac{tan ( 2 x ) tan ( x ) - 1}{tan ( x )} = 0

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

tan (2 x) tan (x) - 1 = 0

Rewrite using trig identities

- 1 + \frac{2 tan ^{2} ( x )}{1 - tan ^{2} ( x )} = 0

Solve by substitution

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3} : x = arctan (\frac{1}{3}) + πn

tan (x) = - \frac{1}{3} : x = arctan (- \frac{1}{3}) + πn

Combine all the solutions

x = arctan (\frac{1}{3}) + πn, x = arctan (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

sin (x) + cos (x) + 1 = 0

tan (x) = 15

4 sin (x) = cos (x) + 2

tan (β + 1 0^{\circ}) = cot (2 β - 1 0^{\circ})

csc^{2} (x) - 2 csc (x) = 2 - 4 sin (x), 0 \leq x \leq 2 π

What is the general solution for tan(2x)-1/(tan(x))=0 ?
The general solution for tan(2x)-1/(tan(x))=0 is x=0.52359…+pin,x=-0.52359…+pin