What is the general solution for (tan(x)(-tan^2(x)+3))/(1-tan^2(x))=0 ?

The general solution for (tan(x)(-tan^2(x)+3))/(1-tan^2(x))=0 is x=pin,x=(2pi)/3+pin,x= pi/3+pin

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(tan(x)(-tan^2(x)+3))/(1-tan^2(x))=0

Solution

\frac{tan ( x ) ( - tan ^{2} ( x ) + 3 )}{1 - tan ^{2} ( x )} = 0

Solution

x = πn, x = \frac{2 π}{3} + πn, x = \frac{π}{3} + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

\frac{tan ( x ) ( - tan ^{2} ( x ) + 3 )}{1 - tan ^{2} ( x )} = 0

Solve by substitution

\frac{tan ( x ) ( - tan ^{2} ( x ) + 3 )}{1 - tan ^{2} ( x )} = 0

Let:

tan (x) = u

\frac{u ( - u ^{2} + 3 )}{1 - u ^{2}} = 0

\frac{u ( - u ^{2} + 3 )}{1 - u ^{2}} = 0 : u = 0, u = - 3, u = 3

\frac{u ( - u ^{2} + 3 )}{1 - u ^{2}} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

u (- u^{2} + 3) = 0

Solve

u (- u^{2} + 3) = 0 : u = 0, u = - 3, u = 3

u (- u^{2} + 3) = 0

Factor

u (- u^{2} + 3) : - u (u + 3) (u - 3)

u (- u^{2} + 3)

Factor

- u^{2} + 3 : - (u + 3) (u - 3)

- u^{2} + 3

Factor out common term

- 1

= - (u^{2} - 3)

Factor

u^{2} - 3 : (u + 3) (u - 3)

u^{2} - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= u^{2} - (3)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - (3)^{2} = (u + 3) (u - 3)

= (u + 3) (u - 3)

= - (u + 3) (u - 3)

= - u (u + 3) (u - 3)

- u (u + 3) (u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u + 3 = 0 or u - 3 = 0

Solve

u + 3 = 0 : u = - 3

u + 3 = 0

Move

3

to the right side

u + 3 = 0

Subtract

3

from both sides

u + 3 - 3 = 0 - 3

Simplify

u = - 3

u = - 3

Solve

u - 3 = 0 : u = 3

u - 3 = 0

Move

3

to the right side

u - 3 = 0

Add

3

to both sides

u - 3 + 3 = 0 + 3

Simplify

u = 3

u = 3

The solutions are

u = 0, u = - 3, u = 3

u = 0, u = - 3, u = 3

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

\frac{u ( - u ^{2} + 3 )}{1 - u ^{2}}

and compare to zero

Solve

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

Move

1

to the right side

1 - u^{2} = 0

Subtract

1

from both sides

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

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 radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u = 0, u = - 3, u = 3

Substitute back

u = tan (x)

tan (x) = 0, tan (x) = - 3, tan (x) = 3

tan (x) = 0, tan (x) = - 3, tan (x) = 3

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

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 = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

tan (x) = - 3 : x = \frac{2 π}{3} + πn

tan (x) = - 3

General solutions for

tan (x) = - 3

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{2 π}{3} + πn

x = \frac{2 π}{3} + πn

tan (x) = 3 : x = \frac{π}{3} + πn

tan (x) = 3

General solutions for

tan (x) = 3

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}{3} + πn

x = \frac{π}{3} + πn

Combine all the solutions

x = πn, x = \frac{2 π}{3} + πn, x = \frac{π}{3} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (tan(x)(-tan^2(x)+3))/(1-tan^2(x))=0 ?
The general solution for (tan(x)(-tan^2(x)+3))/(1-tan^2(x))=0 is x=pin,x=(2pi)/3+pin,x= pi/3+pin