Solve for x tan(x)^4-20tan(x)^2+64=0

Problem

tan(x)−20*tan(x)+64=0

Solution

1. Substitute a new variable to transform the equation into a quadratic form. Let u=tan(x)

u2−20*u+64=0

2. Factor the quadratic equation by finding two numbers that multiply to 64 and add to −20 These numbers are −4 and −16

(u−4)*(u−16)=0

3. Solve for u by setting each factor to zero.

u=4

u=16

4. Back-substitute u=tan(x) to find the values for tan(x)

tan(x)=4⇒tan(x)=±2

tan(x)=16⇒tan(x)=±4

5. Apply the inverse tangent function to solve for x Since the tangent function has a period of π we add n*π where n is an integer.

x=arctan(2)+n*π

x=arctan(−2)+n*π

x=arctan(4)+n*π

x=arctan(−4)+n*π

6. Simplify using the property arctan(−z)=−arctan(z)

x=±arctan(2)+n*π

x=±arctan(4)+n*π

Final Answer

tan(x)−20*tan(x)+64=0⇒x=n*π±arctan(2),n*π±arctan(4)

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