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

The general solution for 1-2tan^2(x)=-tan^2(x) is x= pi/4+pin,x=(3pi)/4+pin

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

Solution

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

Solution

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

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

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Move

u^{2}

to the left side

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

Add

u^{2}

to both sides

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

Simplify

- u^{2} = - 1

- u^{2} = - 1

Divide both sides by

- 1

- u^{2} = - 1

Divide both sides by

- 1

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

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

Substitute back

u = tan (x)

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

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

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

tan (x) = - 1 : x = \frac{3 π}{4} + πn

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Popular Examples

sin(x)=-1/2 ,0<= x<= 2pi

sin (x) = - \frac{1}{2}, 0 \leq x \leq 2 π

sin^2(x)+2-cos^2(x)=3sin(x)

sin^{2} (x) + 2 - cos^{2} (x) = 3 sin (x)

sin^2(a)sec(a)=tan^2(a)

sin^{2} (a) sec (a) = tan^{2} (a)

tan^2(θ)-sec(θ)=1

tan^{2} (θ) - sec (θ) = 1

sin(x)= 8/10

sin (x) = \frac{8}{10}

Frequently Asked Questions (FAQ)

What is the general solution for 1-2tan^2(x)=-tan^2(x) ?
The general solution for 1-2tan^2(x)=-tan^2(x) is x= pi/4+pin,x=(3pi)/4+pin