What is the general solution for tan(u)-cot(u)=2 ?

The general solution for tan(u)-cot(u)=2 is u=2.74889…+pin,u=1.17809…+pin

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tan(u)-cot(u)=2

Solution

tan (u) - cot (u) = 2

Solution

u = 2.74889 \dots + πn, u = 1.17809 \dots + πn

Degrees

u = 157. 5^{\circ} + 18 0^{\circ} n, u = 67. 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (u) - cot (u) = 2

Subtract

2

from both sides

tan (u) - cot (u) - 2 = 0

Rewrite using trig identities

- 2 - cot (u) + tan (u)

Use the basic trigonometric identity:

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

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

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

Solve by substitution

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

Let:

cot (u) = v

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

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

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

Multiply both sides by

v

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

Multiply both sides by

v

- 2 v - vv + \frac{1}{v} v = 0 \cdot v

Simplify

- 2 v - vv + \frac{1}{v} v = 0 \cdot v

Simplify

- vv : - v^{2}

- vv

Apply exponent rule:

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

vv = v^{1 + 1}

= - v^{1 + 1}

Add the numbers:

1 + 1 = 2

= - v^{2}

Simplify

\frac{1}{v} v : 1

\frac{1}{v} v

Multiply fractions:

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

= \frac{1 \cdot v}{v}

Cancel the common factor:

v

= 1

Simplify

0 \cdot v : 0

0 \cdot v

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

- 2 v - v^{2} + 1 = 0 : v = - 1 - 2, v = 2 - 1

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

(- 2)^{2} - 4 (- 1) \cdot 1 = 22

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

Apply rule

- (- a) = a

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 2^{2} + 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2^{2} + 4

2^{2} = 4

= 4 + 4

Add the numbers:

4 + 4 = 8

= 8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

v_{1, 2} = \frac{- ( - 2 ) \pm 2 2}{2 ( - 1 )}

Separate the solutions

v_{1} = \frac{- ( - 2 ) + 2 2}{2 ( - 1 )}, v_{2} = \frac{- ( - 2 ) - 2 2}{2 ( - 1 )}

v = \frac{- ( - 2 ) + 2 2}{2 ( - 1 )} : - 1 - 2

\frac{- ( - 2 ) + 2 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

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

Cancel

\frac{2 + 2 2}{2} : 1 + 2

\frac{2 + 2 2}{2}

Factor

2 + 22 : 2 (1 + 2)

2 + 22

Rewrite as

= 2 \cdot 1 + 22

Factor out common term

2

= 2 (1 + 2)

= \frac{2 ( 1 + 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 2

= - (1 + 2)

Distribute parentheses

= - (1) - (2)

Apply minus-plus rules

+ (- a) = - a

= - 1 - 2

v = \frac{- ( - 2 ) - 2 2}{2 ( - 1 )} : 2 - 1

\frac{- ( - 2 ) - 2 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 - 2 2}{- 2}

Apply the fraction rule:

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

2 - 22 = - (22 - 2)

= \frac{2 2 - 2}{2}

Factor

22 - 2 : 2 (2 - 1)

22 - 2

Rewrite as

= 22 - 2 \cdot 1

Factor out common term

2

= 2 (2 - 1)

= \frac{2 ( 2 - 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 - 1

The solutions to the quadratic equation are:

v = - 1 - 2, v = 2 - 1

v = - 1 - 2, v = 2 - 1

Verify Solutions

Find undefined (singularity) points:

v = 0

Take the denominator(s) of

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

and compare to zero

v = 0

The following points are undefined

v = 0

Combine undefined points with solutions:

v = - 1 - 2, v = 2 - 1

Substitute back

v = cot (u)

cot (u) = - 1 - 2, cot (u) = 2 - 1

cot (u) = - 1 - 2, cot (u) = 2 - 1

cot (u) = - 1 - 2 : u = arccot (- 1 - 2) + πn

cot (u) = - 1 - 2

Apply trig inverse properties

cot (u) = - 1 - 2

General solutions for

cot (u) = - 1 - 2

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

u = arccot (- 1 - 2) + πn

u = arccot (- 1 - 2) + πn

cot (u) = 2 - 1 : u = arccot (2 - 1) + πn

cot (u) = 2 - 1

Apply trig inverse properties

cot (u) = 2 - 1

General solutions for

cot (u) = 2 - 1

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

u = arccot (2 - 1) + πn

u = arccot (2 - 1) + πn

Combine all the solutions

u = arccot (- 1 - 2) + πn, u = arccot (2 - 1) + πn

Show solutions in decimal form

u = 2.74889 \dots + πn, u = 1.17809 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(u)-cot(u)=2 ?
The general solution for tan(u)-cot(u)=2 is u=2.74889…+pin,u=1.17809…+pin