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

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

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tan(5x)tan(x)=tan(5x)

tan (5 x) tan (x) = tan (5 x)

x = \frac{πn}{5}, x = \frac{π}{4} + πn

Solution steps

tan (5 x) tan (x) = tan (5 x)

Subtract

tan (5 x)

from both sides

tan (5 x) tan (x) - tan (5 x) = 0

Factor

tan (5 x) tan (x) - tan (5 x) : tan (5 x) (tan (x) - 1)

tan (5 x) (tan (x) - 1) = 0

Solving each part separately

tan (5 x) = 0 or tan (x) - 1 = 0

tan (5 x) = 0 : x = \frac{πn}{5}

tan (x) - 1 = 0 : x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{πn}{5}, x = \frac{π}{4} + πn

4 sin^{2} (x) = 1 + 4 cos (x)

4 sec (θ) + 23 = sec (θ)

2 cos (3 x) - 3 = 0

cos^{2} (x) - 8 cos (x) - 1 = 0

cos (x) + 2 = 1

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