What is the general solution for 4tan^2(x)-1=tan^2(x) ?

The general solution for 4tan^2(x)-1=tan^2(x) is x=0.52359…+pin,x=-0.52359…+pin

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4tan^2(x)-1=tan^2(x)

Solution

4 tan^{2} (x) - 1 = tan^{2} (x)

Solution

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

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = - 3 0^{\circ} + 18 0^{\circ} n

Solution steps

4 tan^{2} (x) - 1 = tan^{2} (x)

Solve by substitution

4 tan^{2} (x) - 1 = tan^{2} (x)

Let:

tan (x) = u

4 u^{2} - 1 = u^{2}

4 u^{2} - 1 = u^{2} : u = \frac{1}{3}, u = - \frac{1}{3}

4 u^{2} - 1 = u^{2}

Move

1

to the right side

4 u^{2} - 1 = u^{2}

Add

1

to both sides

4 u^{2} - 1 + 1 = u^{2} + 1

Simplify

4 u^{2} = u^{2} + 1

4 u^{2} = u^{2} + 1

Move

u^{2}

to the left side

4 u^{2} = u^{2} + 1

Subtract

u^{2}

from both sides

4 u^{2} - u^{2} = u^{2} + 1 - u^{2}

Simplify

3 u^{2} = 1

3 u^{2} = 1

Divide both sides by

3

3 u^{2} = 1

Divide both sides by

3

\frac{3 u ^{2}}{3} = \frac{1}{3}

Simplify

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = tan (x)

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

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}

Apply trig inverse properties

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

General solutions for

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

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

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

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{1}{3}) + πn

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

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan^2(x)-1=tan^2(x) ?
The general solution for 4tan^2(x)-1=tan^2(x) is x=0.52359…+pin,x=-0.52359…+pin