What is the general solution for (tan(x^2+6)-1)/(tan(x))=0 ?

The general solution for (tan(x^2+6)-1)/(tan(x))=0 is No Solution for x\in\mathbb{R}

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(tan(x^2+6)-1)/(tan(x))=0

Solution

\frac{tan ( x ^{2} + 6 ) - 1}{tan ( x )} = 0

Solution

No Solution for x \in R

Solution steps

\frac{tan ( x ^{2} + 6 ) - 1}{tan ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

tan (x^{2} + 6) - 1 = 0

Move

1

to the right side

tan (x^{2} + 6) - 1 = 0

Add

1

to both sides

tan (x^{2} + 6) - 1 + 1 = 0 + 1

Simplify

tan (x^{2} + 6) = 1

tan (x^{2} + 6) = 1

General solutions for

tan (x^{2} + 6) = 1

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x^{2} + 6 = \frac{π}{4} + πn

x^{2} + 6 = \frac{π}{4} + πn

Solve

x^{2} + 6 = \frac{π}{4} + πn : x = \frac{π + 4 πn - 24}{2}, x = - \frac{π + 4 πn - 24}{2}

x^{2} + 6 = \frac{π}{4} + πn

Move

6

to the right side

x^{2} + 6 = \frac{π}{4} + πn

Subtract

6

from both sides

x^{2} + 6 - 6 = \frac{π}{4} + πn - 6

Simplify

x^{2} = \frac{π}{4} + πn - 6

x^{2} = \frac{π}{4} + πn - 6

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

x = \frac{π}{4} + πn - 6, x = - \frac{π}{4} + πn - 6

Simplify

\frac{π}{4} + πn - 6 : \frac{π + 4 πn - 24}{2}

\frac{π}{4} + πn - 6

Join

\frac{π}{4} + πn - 6 : \frac{π + 4 πn - 24}{4}

\frac{π}{4} + πn - 6

Convert element to fraction:

πn = \frac{πn 4}{4}, 6 = \frac{6 \cdot 4}{4}

= \frac{π}{4} + \frac{πn \cdot 4}{4} - \frac{6 \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π + πn \cdot 4 - 6 \cdot 4}{4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{π + 4 πn - 24}{4}

= \frac{π + πn \cdot 4 - 24}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{π + 4 πn - 24}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{π + 4 πn - 24}{2}

Simplify

- \frac{π}{4} + πn - 6 : - \frac{π + 4 πn - 24}{2}

- \frac{π}{4} + πn - 6

Join

\frac{π}{4} + πn - 6 : \frac{π + 4 πn - 24}{4}

\frac{π}{4} + πn - 6

Convert element to fraction:

πn = \frac{πn 4}{4}, 6 = \frac{6 \cdot 4}{4}

= \frac{π}{4} + \frac{πn \cdot 4}{4} - \frac{6 \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π + πn \cdot 4 - 6 \cdot 4}{4}

Multiply the numbers:

6 \cdot 4 = 24

= \frac{π + 4 πn - 24}{4}

= - \frac{π + 4 πn - 24}{4}

Simplify

\frac{π + πn \cdot 4 - 24}{4} : \frac{π + 4 πn - 24}{2}

\frac{π + πn \cdot 4 - 24}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{π + 4 πn - 24}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{π + 4 πn - 24}{2}

= - \frac{π + 4 πn - 24}{2}

x = \frac{π + 4 πn - 24}{2}, x = - \frac{π + 4 πn - 24}{2}

x = \frac{π + 4 πn - 24}{2}, x = - \frac{π + 4 πn - 24}{2}

Since the equation is undefined for:

\frac{π + 4 πn - 24}{2}, - \frac{π + 4 πn - 24}{2}

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (tan(x^2+6)-1)/(tan(x))=0 ?
The general solution for (tan(x^2+6)-1)/(tan(x))=0 is No Solution for x\in\mathbb{R}