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

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

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tan(2x)tan(x)=1

tan (2 x) tan (x) = 1

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

Solution steps

tan (2 x) tan (x) = 1

Subtract

1

from both sides

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

\frac{csc ^{2} ( x )}{4} = 4 sin^{2} (x)

6 sec^{2} (x) - 3 cos (x) - 10 = sec (x)

1 + 4 cos (θ) = 3 sin (θ)

tan (x) - sec (x) = 3

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

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