What is the general solution for tan(x)tan(7x)=1,x ?

The general solution for tan(x)tan(7x)=1,x is No Solution for x\in\mathbb{R}

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tan(x)tan(7x)=1,x

Solution

tan (x) tan (7 x) = 1, x

Solution

No Solution for x \in R

Solution steps

tan (x) tan (7 x) = 1, x

Subtract

1

from both sides

tan (x) tan (7 x) - 1 = 0

Express with sin, cos

- 1 + tan (7 x) tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 1 + \frac{sin ( 7 x )}{cos ( 7 x )} tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 1 + \frac{sin ( 7 x )}{cos ( 7 x )} \cdot \frac{sin ( x )}{cos ( x )}

Simplify

- 1 + \frac{sin ( 7 x )}{cos ( 7 x )} \cdot \frac{sin ( x )}{cos ( x )} : \frac{- cos ( 7 x ) cos ( x ) + sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

- 1 + \frac{sin ( 7 x )}{cos ( 7 x )} \cdot \frac{sin ( x )}{cos ( x )}

Multiply

\frac{sin ( 7 x )}{cos ( 7 x )} \cdot \frac{sin ( x )}{cos ( x )} : \frac{sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

\frac{sin ( 7 x )}{cos ( 7 x )} \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

= - 1 + \frac{sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

Convert element to fraction:

1 = \frac{1 cos ( 7 x ) cos ( x )}{cos ( 7 x ) cos ( x )}

= - \frac{1 \cdot cos ( 7 x ) cos ( x )}{cos ( 7 x ) cos ( x )} + \frac{sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot cos ( 7 x ) cos ( x ) + sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

Multiply:

1 \cdot cos (7 x) = cos (7 x)

= \frac{- cos ( 7 x ) cos ( x ) + sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

= \frac{- cos ( 7 x ) cos ( x ) + sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )}

\frac{- cos ( 7 x ) cos ( x ) + sin ( 7 x ) sin ( x )}{cos ( 7 x ) cos ( x )} = 0

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

- cos (7 x) cos (x) + sin (7 x) sin (x) = 0

Rewrite using trig identities

- cos (7 x) cos (x) + sin (7 x) sin (x)

Use the Angle Sum identity:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

- cos (s) cos (t) + sin (s) sin (t) = - cos (s + t)

= - cos (7 x + x)

- cos (7 x + x) = 0

Divide both sides by

- 1

- cos (7 x + x) = 0

Divide both sides by

- 1

\frac{- cos ( 7 x + x )}{- 1} = \frac{0}{- 1}

Simplify

cos (7 x + x) = 0

cos (7 x + x) = 0

General solutions for

cos (7 x + x) = 0

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

7 x + x = \frac{π}{2} + 2 πn, 7 x + x = \frac{3 π}{2} + 2 πn

7 x + x = \frac{π}{2} + 2 πn, 7 x + x = \frac{3 π}{2} + 2 πn

Solve

7 x + x = \frac{π}{2} + 2 πn : x = \frac{π}{16} + \frac{πn}{4}

7 x + x = \frac{π}{2} + 2 πn

Add similar elements:

7 x + x = 8 x

8 x = \frac{π}{2} + 2 πn

Divide both sides by

8

8 x = \frac{π}{2} + 2 πn

Divide both sides by

8

\frac{8 x}{8} = \frac{\frac{π}{2}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} = \frac{\frac{π}{2}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{\frac{π}{2}}{8} + \frac{2 πn}{8} : \frac{π}{16} + \frac{πn}{4}

\frac{\frac{π}{2}}{8} + \frac{2 πn}{8}

\frac{\frac{π}{2}}{8} = \frac{π}{16}

\frac{\frac{π}{2}}{8}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{π}{16}

\frac{2 πn}{8} = \frac{πn}{4}

\frac{2 πn}{8}

Cancel the common factor:

2

= \frac{πn}{4}

= \frac{π}{16} + \frac{πn}{4}

x = \frac{π}{16} + \frac{πn}{4}

x = \frac{π}{16} + \frac{πn}{4}

x = \frac{π}{16} + \frac{πn}{4}

Solve

7 x + x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{16} + \frac{πn}{4}

7 x + x = \frac{3 π}{2} + 2 πn

Add similar elements:

7 x + x = 8 x

8 x = \frac{3 π}{2} + 2 πn

Divide both sides by

8

8 x = \frac{3 π}{2} + 2 πn

Divide both sides by

8

\frac{8 x}{8} = \frac{\frac{3 π}{2}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} = \frac{\frac{3 π}{2}}{8} + \frac{2 πn}{8}

Simplify

\frac{8 x}{8} : x

\frac{8 x}{8}

Divide the numbers:

\frac{8}{8} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{8} + \frac{2 πn}{8} : \frac{3 π}{16} + \frac{πn}{4}

\frac{\frac{3 π}{2}}{8} + \frac{2 πn}{8}

\frac{\frac{3 π}{2}}{8} = \frac{3 π}{16}

\frac{\frac{3 π}{2}}{8}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{3 π}{16}

\frac{2 πn}{8} = \frac{πn}{4}

\frac{2 πn}{8}

Cancel the common factor:

2

= \frac{πn}{4}

= \frac{3 π}{16} + \frac{πn}{4}

x = \frac{3 π}{16} + \frac{πn}{4}

x = \frac{3 π}{16} + \frac{πn}{4}

x = \frac{3 π}{16} + \frac{πn}{4}

x = \frac{π}{16} + \frac{πn}{4}, x = \frac{3 π}{16} + \frac{πn}{4}

Solutions for the range

x

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)tan(7x)=1,x ?
The general solution for tan(x)tan(7x)=1,x is No Solution for x\in\mathbb{R}