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

The general solution for 1-2tan(x)=tan^2(x) is x=0.39269…+pin,x=-1.17809…+pin

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

Solution

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

Solution

x = 0.39269 \dots + πn, x = - 1.17809 \dots + πn

Degrees

x = 22. 5^{\circ} + 18 0^{\circ} n, x = - 67. 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

1 - 2 u = u^{2}

1 - 2 u = u^{2} : u = - 1 + 2, u = - 1 - 2

1 - 2 u = u^{2}

Switch sides

u^{2} = 1 - 2 u

Move

2 u

to the left side

u^{2} = 1 - 2 u

Add

2 u

to both sides

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

Simplify

u^{2} + 2 u = 1

u^{2} + 2 u = 1

Move

1

to the left side

u^{2} + 2 u = 1

Subtract

1

from both sides

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

Simplify

u^{2} + 2 u - 1 = 0

u^{2} + 2 u - 1 = 0

Solve with the quadratic formula

u^{2} + 2 u - 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 2, c = - 1

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

2^{2} - 4 \cdot 1 \cdot (- 1) = 22

2^{2} - 4 \cdot 1 \cdot (- 1)

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2^{2} + 4

2^{2} = 4

= 4 + 4

Add the numbers:

4 + 4 = 8

= 8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

u_{1, 2} = \frac{- 2 \pm 2 2}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 2 + 2 2}{2 \cdot 1}, u_{2} = \frac{- 2 - 2 2}{2 \cdot 1}

u = \frac{- 2 + 2 2}{2 \cdot 1} : - 1 + 2

\frac{- 2 + 2 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2 + 2 2}{2}

Factor

- 2 + 22 : 2 (- 1 + 2)

- 2 + 22

Rewrite as

= - 2 \cdot 1 + 22

Factor out common term

2

= 2 (- 1 + 2)

= \frac{2 ( - 1 + 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 1 + 2

u = \frac{- 2 - 2 2}{2 \cdot 1} : - 1 - 2

\frac{- 2 - 2 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2 - 2 2}{2}

Factor

- 2 - 22 : - 2 (1 + 2)

- 2 - 22

Rewrite as

= - 2 \cdot 1 - 22

Factor out common term

2

= - 2 (1 + 2)

= - \frac{2 ( 1 + 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - (1 + 2)

Negate

- (1 + 2) = - 1 - 2

= - 1 - 2

The solutions to the quadratic equation are:

u = - 1 + 2, u = - 1 - 2

Substitute back

u = tan (x)

tan (x) = - 1 + 2, tan (x) = - 1 - 2

tan (x) = - 1 + 2, tan (x) = - 1 - 2

tan (x) = - 1 + 2 : x = arctan (- 1 + 2) + πn

tan (x) = - 1 + 2

Apply trig inverse properties

tan (x) = - 1 + 2

General solutions for

tan (x) = - 1 + 2

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

x = arctan (- 1 + 2) + πn

x = arctan (- 1 + 2) + πn

tan (x) = - 1 - 2 : x = arctan (- 1 - 2) + πn

tan (x) = - 1 - 2

Apply trig inverse properties

tan (x) = - 1 - 2

General solutions for

tan (x) = - 1 - 2

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

x = arctan (- 1 - 2) + πn

x = arctan (- 1 - 2) + πn

Combine all the solutions

x = arctan (- 1 + 2) + πn, x = arctan (- 1 - 2) + πn

Show solutions in decimal form

x = 0.39269 \dots + πn, x = - 1.17809 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 1-2tan(x)=tan^2(x) ?
The general solution for 1-2tan(x)=tan^2(x) is x=0.39269…+pin,x=-1.17809…+pin