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tan^2(x)+tan(x)+cot(x)+cot^2(x)=4

Solution

tan^{2} (x) + tan (x) + cot (x) + cot^{2} (x) = 4

Solution

x = \frac{π}{4} + πn, x = 1.93566 \dots + πn, x = 2.77672 \dots + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 110.90515 \dots^{\circ} + 18 0^{\circ} n, x = 159.09484 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) + tan (x) + cot (x) + cot^{2} (x) = 4

Subtract

4

from both sides

tan^{2} (x) + tan (x) + cot (x) + cot^{2} (x) - 4 = 0

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= - 4 + cot (x) + cot^{2} (x) + \frac{1}{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 )}

= - 4 + cot (x) + cot^{2} (x) + \frac{1}{cot ( x )} + \frac{1}{cot ^{2} ( x )}

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

Solve by substitution

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

Let:

cot (x) = u

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

- 4 + u + u^{2} + \frac{1}{u ^{2}} + \frac{1}{u} = 0 : u = 1, u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

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

Multiply by LCM

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

Find Least Common Multiplier of

u^{2}, u : u^{2}

u^{2}, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u^{2}

u

= u^{2}

Multiply by LCM=

u^{2}

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

Simplify

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

u^{2} u^{2} : u^{4}

u^{2} u^{2}

Apply exponent rule:

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

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

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

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

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

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

u

= u

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

- 4 u^{2} + u^{3} + u^{4} + 1 + u = 0

- 4 u^{2} + u^{3} + u^{4} + 1 + u = 0

- 4 u^{2} + u^{3} + u^{4} + 1 + u = 0

Solve

- 4 u^{2} + u^{3} + u^{4} + 1 + u = 0 : u = 1, u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

- 4 u^{2} + u^{3} + u^{4} + 1 + u = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} + u^{3} - 4 u^{2} + u + 1 = 0

Factor

u^{4} + u^{3} - 4 u^{2} + u + 1 : (u - 1)^{2} (u^{2} + 3 u + 1)

u^{4} + u^{3} - 4 u^{2} + u + 1

Use the rational root theorem

a_{0} = 1, a_{n} = 1

The dividers of

a_{0} : 1,

The dividers of

a_{n} : 1

Therefore, check the following rational numbers:

\pm \frac{1}{1}

\frac{1}{1}

is a root of the expression, so factor out

u - 1

= (u - 1) \frac{u ^{4} + u ^{3} - 4 u ^{2} + u + 1}{u - 1}

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

\frac{u ^{4} + u ^{3} - 4 u ^{2} + u + 1}{u - 1}

Divide

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

Divide the leading coefficients of the numerator

u^{4} + u^{3} - 4 u^{2} + u + 1

and the divisor

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

Quotient = u^{3}

Multiply

u - 1

u^{3} : u^{4} - u^{3}

Subtract

u^{4} - u^{3}

from

u^{4} + u^{3} - 4 u^{2} + u + 1

to get new remainder

Remainder = 2 u^{3} - 4 u^{2} + u + 1

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

2 u^{3} - 4 u^{2} + u + 1

and the divisor

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

Quotient = 2 u^{2}

Multiply

u - 1

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

Subtract

2 u^{3} - 2 u^{2}

from

2 u^{3} - 4 u^{2} + u + 1

to get new remainder

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

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

- 2 u^{2} + u + 1

and the divisor

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

Quotient = - 2 u

Multiply

u - 1

- 2 u : - 2 u^{2} + 2 u

Subtract

- 2 u^{2} + 2 u

from

- 2 u^{2} + u + 1

to get new remainder

Remainder = - u + 1

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

- u + 1

and the divisor

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

Quotient = - 1

Multiply

u - 1

- 1 : - u + 1

Subtract

- u + 1

from

- u + 1

to get new remainder

Remainder = 0

Therefore

\frac{- u + 1}{u - 1} = - 1

= u^{3} + 2 u^{2} - 2 u - 1

= u^{3} + 2 u^{2} - 2 u - 1

Factor

u^{3} + 2 u^{2} - 2 u - 1 : (u - 1) (u^{2} + 3 u + 1)

u^{3} + 2 u^{2} - 2 u - 1

Use the rational root theorem

a_{0} = 1, a_{n} = 1

The dividers of

a_{0} : 1,

The dividers of

a_{n} : 1

Therefore, check the following rational numbers:

\pm \frac{1}{1}

\frac{1}{1}

is a root of the expression, so factor out

u - 1

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

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

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

Divide

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

Divide the leading coefficients of the numerator

u^{3} + 2 u^{2} - 2 u - 1

and the divisor

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

Quotient = u^{2}

Multiply

u - 1

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

Subtract

u^{3} - u^{2}

from

u^{3} + 2 u^{2} - 2 u - 1

to get new remainder

Remainder = 3 u^{2} - 2 u - 1

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

3 u^{2} - 2 u - 1

and the divisor

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

Quotient = 3 u

Multiply

u - 1

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

Subtract

3 u^{2} - 3 u

from

3 u^{2} - 2 u - 1

to get new remainder

Remainder = u - 1

Therefore

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

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

Divide

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

Divide the leading coefficients of the numerator

u - 1

and the divisor

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

Quotient = 1

Multiply

u - 1

1 : u - 1

Subtract

u - 1

from

u - 1

to get new remainder

Remainder = 0

Therefore

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

= u^{2} + 3 u + 1

= u^{2} + 3 u + 1

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

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

Refine

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u - 1 = 0 or u^{2} + 3 u + 1 = 0

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

u^{2} + 3 u + 1 = 0 : u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

u^{2} + 3 u + 1 = 0

Solve with the quadratic formula

u^{2} + 3 u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = 1

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

3^{2} - 4 \cdot 1 \cdot 1 = 5

3^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 3^{2} - 4

3^{2} = 9

= 9 - 4

Subtract the numbers:

9 - 4 = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 \cdot 1}, u_{2} = \frac{- 3 - 5}{2 \cdot 1}

u = \frac{- 3 + 5}{2 \cdot 1} : \frac{- 3 + 5}{2}

\frac{- 3 + 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 + 5}{2}

u = \frac{- 3 - 5}{2 \cdot 1} : \frac{- 3 - 5}{2}

\frac{- 3 - 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 - 5}{2}

The solutions to the quadratic equation are:

u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

The solutions are

u = 1, u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

u = 1, u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

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

u = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

Substitute back

u = cot (x)

cot (x) = 1, cot (x) = \frac{- 3 + 5}{2}, cot (x) = \frac{- 3 - 5}{2}

cot (x) = 1, cot (x) = \frac{- 3 + 5}{2}, cot (x) = \frac{- 3 - 5}{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{- 3 + 5}{2} : x = arccot (\frac{- 3 + 5}{2}) + πn

cot (x) = \frac{- 3 + 5}{2}

Apply trig inverse properties

cot (x) = \frac{- 3 + 5}{2}

General solutions for

cot (x) = \frac{- 3 + 5}{2}

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (\frac{- 3 + 5}{2}) + πn

x = arccot (\frac{- 3 + 5}{2}) + πn

cot (x) = \frac{- 3 - 5}{2} : x = arccot (\frac{- 3 - 5}{2}) + πn

cot (x) = \frac{- 3 - 5}{2}

Apply trig inverse properties

cot (x) = \frac{- 3 - 5}{2}

General solutions for

cot (x) = \frac{- 3 - 5}{2}

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (\frac{- 3 - 5}{2}) + πn

x = arccot (\frac{- 3 - 5}{2}) + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = arccot (\frac{- 3 + 5}{2}) + πn, x = arccot (\frac{- 3 - 5}{2}) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = 1.93566 \dots + πn, x = 2.77672 \dots + πn

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