What is the general solution for tan(x)+cot(x)=3,=tan(3x)+cot(3x) ?

The general solution for tan(x)+cot(x)=3,=tan(3x)+cot(3x) is No Solution for x\in\mathbb{R}

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tan(x)+cot(x)=3,=tan(3x)+cot(3x)

Solution

tan (x) + cot (x) = 3, = tan (3 x) + cot (3 x)

Solution

No Solution for x \in R

Solution steps

tan (x) + cot (x) = 3, = tan (3 x) + cot (3 x)

Subtract

3

from both sides

tan (x) + cot (x) - 3 = 0

Rewrite using trig identities

- 3 + cot (x) + tan (x)

Use the basic trigonometric identity:

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

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

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

Solve by substitution

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

Let:

cot (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

uu : u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

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

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

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

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

Write in the standard form

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

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

Apply exponent rule:

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

n

is even

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

= 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}

Apply rule

- (- a) = a

= \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}

Apply rule

- (- a) = a

= \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}

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = cot (x)

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

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

cot (x) = \frac{3 + 5}{2}, = tan (3 x) + cot (3 x) :

No Solution

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

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

Solutions for the range

= tan (3 x) + cot (3 x)

No Solution

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

No Solution

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

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

Solutions for the range

= tan (3 x) + cot (3 x)

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)+cot(x)=3,=tan(3x)+cot(3x) ?
The general solution for tan(x)+cot(x)=3,=tan(3x)+cot(3x) is No Solution for x\in\mathbb{R}