What is the general solution for tan(x)cot(x)-tan(x)+2cot(x)=0 ?

The general solution for tan(x)cot(x)-tan(x)+2cot(x)=0 is x=1.10714…+pin,x=(3pi)/4+pin

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tan(x)cot(x)-tan(x)+2cot(x)=0

Solution

tan (x) cot (x) - tan (x) + 2 cot (x) = 0

Solution

x = 1.10714 \dots + πn, x = \frac{3 π}{4} + πn

Degrees

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) cot (x) - tan (x) + 2 cot (x) = 0

Rewrite using trig identities

- tan (x) + 2 cot (x) + cot (x) tan (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cot (x)

= 1

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

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

Solve by substitution

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

Let:

cot (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

1 \cdot u : u

1 \cdot u

Multiply:

1 \cdot u = u

= u

Simplify

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

- \frac{1}{u} u

Multiply fractions:

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

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

Cancel the common factor:

u

= - 1

Simplify

2 uu : 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Write in the standard form

a x^{2} + b x + c = 0

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

1^{2} - 4 \cdot 2 (- 1) = 3

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 1)

Apply rule

- (- a) = a

= 1 + 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 1 + 8

Add the numbers:

1 + 8 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

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

Separate the solutions

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

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

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

Add/Subtract the numbers:

- 1 + 3 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

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

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

Subtract the numbers:

- 1 - 3 = - 4

= \frac{- 4}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = cot (x)

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

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

cot (x) = \frac{1}{2} : x = arccot (\frac{1}{2}) + πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arccot (\frac{1}{2}) + πn

x = arccot (\frac{1}{2}) + πn

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

x = \frac{3 π}{4} + πn

Combine all the solutions

x = arccot (\frac{1}{2}) + πn, x = \frac{3 π}{4} + πn

Show solutions in decimal form

x = 1.10714 \dots + πn, x = \frac{3 π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)cot(x)-tan(x)+2cot(x)=0 ?
The general solution for tan(x)cot(x)-tan(x)+2cot(x)=0 is x=1.10714…+pin,x=(3pi)/4+pin