What is the general solution for tan(x+20)*tan(x-20)=1 ?

The general solution for tan(x+20)*tan(x-20)=1 is x=45+180n,x=135+180n

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tan(x+20)*tan(x-20)=1

Solution

tan (x + 2 0^{\circ}) \cdot tan (x - 2 0^{\circ}) = 1

Solution

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Radians

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

Solution steps

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

Subtract

1

from both sides

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

Simplify

tan (x + 2 0^{\circ}) tan (x - 2 0^{\circ}) - 1 : tan (\frac{9 x + 18 0 ^{\circ}}{9}) tan (\frac{9 x - 18 0 ^{\circ}}{9}) - 1

tan (x + 2 0^{\circ}) tan (x - 2 0^{\circ}) - 1

tan (x + 2 0^{\circ}) tan (x - 2 0^{\circ}) = tan (\frac{9 x + 18 0 ^{\circ}}{9}) tan (\frac{9 x - 18 0 ^{\circ}}{9})

tan (x + 2 0^{\circ}) tan (x - 2 0^{\circ})

Join

x + 2 0^{\circ} : \frac{9 x + 18 0 ^{\circ}}{9}

x + 2 0^{\circ}

Convert element to fraction:

x = \frac{x 9}{9}

= \frac{x \cdot 9}{9} + 2 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 9 + 18 0 ^{\circ}}{9}

= tan (\frac{9 x + 18 0 ^{\circ}}{9}) tan (x - 2 0^{\circ})

Join

x - 2 0^{\circ} : \frac{9 x - 18 0 ^{\circ}}{9}

x - 2 0^{\circ}

Convert element to fraction:

x = \frac{x 9}{9}

= \frac{x \cdot 9}{9} - 2 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 9 - 18 0 ^{\circ}}{9}

= tan (\frac{9 x + 18 0 ^{\circ}}{9}) tan (\frac{9 x - 18 0 ^{\circ}}{9})

= tan (\frac{9 x + 18 0 ^{\circ}}{9}) tan (\frac{9 x - 18 0 ^{\circ}}{9}) - 1

tan (\frac{9 x + 18 0 ^{\circ}}{9}) tan (\frac{9 x - 18 0 ^{\circ}}{9}) - 1 = 0

Express with sin, cos

- 1 + tan (\frac{- 18 0 ^{\circ} + 9 x}{9}) tan (\frac{18 0 ^{\circ} + 9 x}{9})

Use the basic trigonometric identity:

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

= - 1 + \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )} tan (\frac{18 0 ^{\circ} + 9 x}{9})

Use the basic trigonometric identity:

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

= - 1 + \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )} \cdot \frac{sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

Simplify

- 1 + \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )} \cdot \frac{sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} )} : \frac{- cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) + sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

- 1 + \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )} \cdot \frac{sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

Multiply

\frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )} \cdot \frac{sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} )} : \frac{sin ( \frac{9 x - 18 0 ^{\circ}}{9} ) sin ( \frac{9 x + 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x - 18 0 ^{\circ}}{9} ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

\frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} )} \cdot \frac{sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

Multiply fractions:

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

= \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

= - 1 + \frac{sin ( \frac{9 x - 18 0 ^{\circ}}{9} ) sin ( \frac{9 x + 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x - 18 0 ^{\circ}}{9} ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

Convert element to fraction:

1 = \frac{1 cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

= - \frac{1 \cdot cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )} + \frac{sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) + sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

Multiply:

1 \cdot cos (\frac{- 18 0 ^{\circ} + 9 x}{9}) = cos (\frac{- 18 0 ^{\circ} + 9 x}{9})

= \frac{- cos ( \frac{9 x - 18 0 ^{\circ}}{9} ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} ) + sin ( \frac{9 x - 18 0 ^{\circ}}{9} ) sin ( \frac{9 x + 18 0 ^{\circ}}{9} )}{cos ( \frac{9 x - 18 0 ^{\circ}}{9} ) cos ( \frac{9 x + 18 0 ^{\circ}}{9} )}

= \frac{- cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) + sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )}

\frac{- cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} ) + sin ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) sin ( \frac{18 0 ^{\circ} + 9 x}{9} )}{cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} ) cos ( \frac{18 0 ^{\circ} + 9 x}{9} )} = 0

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

- cos (\frac{- 18 0 ^{\circ} + 9 x}{9}) cos (\frac{18 0 ^{\circ} + 9 x}{9}) + sin (\frac{- 18 0 ^{\circ} + 9 x}{9}) sin (\frac{18 0 ^{\circ} + 9 x}{9}) = 0

Rewrite using trig identities

- cos (\frac{- 18 0 ^{\circ} + 9 x}{9}) cos (\frac{18 0 ^{\circ} + 9 x}{9}) + sin (\frac{- 18 0 ^{\circ} + 9 x}{9}) sin (\frac{18 0 ^{\circ} + 9 x}{9})

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 (\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9})

- cos (\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}) = 0

Divide both sides by

- 1

- cos (\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}) = 0

Divide both sides by

- 1

\frac{- cos ( \frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} )}{- 1} = \frac{0}{- 1}

Simplify

cos (\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}) = 0

cos (\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}) = 0

General solutions for

cos (\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 27 0^{\circ} + 36 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 9 0^{\circ} + 36 0^{\circ} n, \frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 27 0^{\circ} + 36 0^{\circ} n

Solve

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 9 0^{\circ} + 36 0^{\circ} n : x = 4 5^{\circ} + 18 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 9 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 + \frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 = 9 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 + \frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 = 9 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 : - 18 0^{\circ} + 9 x

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

Cancel the common factor:

9

= - - 18 0^{\circ} + 9 x

Simplify

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 : 18 0^{\circ} + 9 x

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

= \frac{( 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

Cancel the common factor:

9

= 18 0^{\circ} + 9 x

Simplify

9 0^{\circ} \cdot 9 : 81 0^{\circ}

9 0^{\circ} \cdot 9

Multiply fractions:

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

= 81 0^{\circ}

Simplify

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

Multiply the numbers:

2 \cdot 9 = 18

= 324 0^{\circ} n

- 18 0^{\circ} + 9 x + 18 0^{\circ} + 9 x = 81 0^{\circ} + 324 0^{\circ} n

18 x = 81 0^{\circ} + 324 0^{\circ} n

18 x = 81 0^{\circ} + 324 0^{\circ} n

18 x = 81 0^{\circ} + 324 0^{\circ} n

Divide both sides by

18

18 x = 81 0^{\circ} + 324 0^{\circ} n

Divide both sides by

18

\frac{18 x}{18} = \frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18}

Simplify

\frac{18 x}{18} = \frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18}

Simplify

\frac{18 x}{18} : x

\frac{18 x}{18}

Divide the numbers:

\frac{18}{18} = 1

= x

Simplify

\frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} : 4 5^{\circ} + 18 0^{\circ} n

\frac{81 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18}

\frac{81 0 ^{\circ}}{18} = 4 5^{\circ}

\frac{81 0 ^{\circ}}{18}

Apply the fraction rule:

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

= \frac{162 0 ^{\circ}}{2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= 4 5^{\circ}

Cancel the common factor:

9

= 4 5^{\circ}

\frac{324 0 ^{\circ} n}{18} = 18 0^{\circ} n

\frac{324 0 ^{\circ} n}{18}

Divide the numbers:

\frac{18}{18} = 1

= 18 0^{\circ} n

= 4 5^{\circ} + 18 0^{\circ} n

x = 4 5^{\circ} + 18 0^{\circ} n

x = 4 5^{\circ} + 18 0^{\circ} n

x = 4 5^{\circ} + 18 0^{\circ} n

Solve

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 27 0^{\circ} + 36 0^{\circ} n : x = 13 5^{\circ} + 18 0^{\circ} n

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9} = 27 0^{\circ} + 36 0^{\circ} n

Multiply both sides by

9

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 + \frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 = 27 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 + \frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 = 27 0^{\circ} \cdot 9 + 36 0^{\circ} n \cdot 9

Simplify

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9 : - 18 0^{\circ} + 9 x

\frac{- 18 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

= \frac{( - 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

Cancel the common factor:

9

= - - 18 0^{\circ} + 9 x

Simplify

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9 : 18 0^{\circ} + 9 x

\frac{18 0 ^{\circ} + 9 x}{9} \cdot 9

Multiply fractions:

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

= \frac{( 18 0 ^{\circ} + 9 x ) \cdot 9}{9}

Cancel the common factor:

9

= 18 0^{\circ} + 9 x

Simplify

27 0^{\circ} \cdot 9 : 243 0^{\circ}

27 0^{\circ} \cdot 9

Multiply fractions:

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

= 243 0^{\circ}

Multiply the numbers:

3 \cdot 9 = 27

= 243 0^{\circ}

Simplify

36 0^{\circ} n \cdot 9 : 324 0^{\circ} n

36 0^{\circ} n \cdot 9

Multiply the numbers:

2 \cdot 9 = 18

= 324 0^{\circ} n

- 18 0^{\circ} + 9 x + 18 0^{\circ} + 9 x = 243 0^{\circ} + 324 0^{\circ} n

18 x = 243 0^{\circ} + 324 0^{\circ} n

18 x = 243 0^{\circ} + 324 0^{\circ} n

18 x = 243 0^{\circ} + 324 0^{\circ} n

Divide both sides by

18

18 x = 243 0^{\circ} + 324 0^{\circ} n

Divide both sides by

18

\frac{18 x}{18} = \frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18}

Simplify

\frac{18 x}{18} = \frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18}

Simplify

\frac{18 x}{18} : x

\frac{18 x}{18}

Divide the numbers:

\frac{18}{18} = 1

= x

Simplify

\frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18} : 13 5^{\circ} + 18 0^{\circ} n

\frac{243 0 ^{\circ}}{18} + \frac{324 0 ^{\circ} n}{18}

\frac{243 0 ^{\circ}}{18} = 13 5^{\circ}

\frac{243 0 ^{\circ}}{18}

Apply the fraction rule:

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

= \frac{486 0 ^{\circ}}{2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= 13 5^{\circ}

Cancel the common factor:

9

= 13 5^{\circ}

\frac{324 0 ^{\circ} n}{18} = 18 0^{\circ} n

\frac{324 0 ^{\circ} n}{18}

Divide the numbers:

\frac{18}{18} = 1

= 18 0^{\circ} n

= 13 5^{\circ} + 18 0^{\circ} n

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

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

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

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x+20)*tan(x-20)=1 ?
The general solution for tan(x+20)*tan(x-20)=1 is x=45+180n,x=135+180n