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

The general solution for cot(x)=tan^2(x) is x= pi/4+pin

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cot(x)=tan^2(x)

Solution

cot (x) = tan^{2} (x)

Solution

x = \frac{π}{4} + πn

Degrees

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

Solution steps

cot (x) = tan^{2} (x)

Subtract

tan^{2} (x)

from both sides

cot (x) - tan^{2} (x) = 0

Rewrite using trig identities

cot (x) - tan^{2} (x)

Use the basic trigonometric identity:

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

= cot (x) - (\frac{1}{cot ( x )})^{2}

(\frac{1}{cot ( x )})^{2} = \frac{1}{cot ^{2} ( x )}

(\frac{1}{cot ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cot ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( x )}

= cot (x) - \frac{1}{cot ^{2} ( x )}

cot (x) - \frac{1}{cot ^{2} ( x )} = 0

Solve by substitution

cot (x) - \frac{1}{cot ^{2} ( x )} = 0

Let:

cot (x) = u

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

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

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

Multiply both sides by

u^{2}

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

Multiply both sides by

u^{2}

u u^{2} - \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

u u^{2} - \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplify

u u^{2} : u^{3}

u u^{2}

Apply exponent rule:

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

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

= u^{1 + 2}

Add the numbers:

1 + 2 = 3

= u^{3}

Simplify

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

u^{2}

= - 1

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

u^{3} - 1 = 0

u^{3} - 1 = 0

u^{3} - 1 = 0

Solve

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

u^{3} - 1 = 0

Move

1

to the right side

u^{3} - 1 = 0

Add

1

to both sides

u^{3} - 1 + 1 = 0 + 1

Simplify

u^{3} = 1

u^{3} = 1

For

x^{3} = f (a)

the solutions are

u = 1, u = \frac{- 1 + 3 i}{2}, u = \frac{- 1 - 3 i}{2}

Simplify

\frac{- 1 + 3 i}{2} : - \frac{1}{2} + i \frac{3}{2}

\frac{- 1 + 3 i}{2}

Rewrite

\frac{- 1 + 3 i}{2}

in standard complex form:

- \frac{1}{2} + \frac{3}{2} i

\frac{- 1 + 3 i}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 + 3 i}{2} = - \frac{1}{2} + \frac{3 i}{2}

= - \frac{1}{2} + \frac{3 i}{2}

= - \frac{1}{2} + \frac{3}{2} i

Simplify

\frac{- 1 - 3 i}{2} : - \frac{1}{2} - i \frac{3}{2}

\frac{- 1 - 3 i}{2}

Rewrite

\frac{- 1 - 3 i}{2}

in standard complex form:

- \frac{1}{2} - \frac{3}{2} i

\frac{- 1 - 3 i}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 - 3 i}{2} = - \frac{1}{2} - \frac{3 i}{2}

= - \frac{1}{2} - \frac{3 i}{2}

= - \frac{1}{2} - \frac{3}{2} i

u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - \frac{1}{2} + i \frac{3}{2}, u = - \frac{1}{2} - i \frac{3}{2}

Substitute back

u = cot (x)

cot (x) = 1, cot (x) = - \frac{1}{2} + i \frac{3}{2}, cot (x) = - \frac{1}{2} - i \frac{3}{2}

cot (x) = 1, cot (x) = - \frac{1}{2} + i \frac{3}{2}, cot (x) = - \frac{1}{2} - i \frac{3}{2}

cot (x) = 1 : x = \frac{π}{4} + πn

cot (x) = 1

General solutions for

cot (x) = 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

cot (x) = - \frac{1}{2} + i \frac{3}{2} :

No Solution

cot (x) = - \frac{1}{2} + i \frac{3}{2}

No Solution

cot (x) = - \frac{1}{2} - i \frac{3}{2} :

No Solution

cot (x) = - \frac{1}{2} - i \frac{3}{2}

No Solution

Combine all the solutions

x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

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