What is the general solution for tan(2x)+sec(2x)=1 ?

The general solution for tan(2x)+sec(2x)=1 is x=pin+pi

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tan(2x)+sec(2x)=1

Solution

tan (2 x) + sec (2 x) = 1

Solution

x = πn + π

Degrees

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

Solution steps

tan (2 x) + sec (2 x) = 1

Subtract

1

from both sides

tan (2 x) + sec (2 x) - 1 = 0

Express with sin, cos

- 1 + sec (2 x) + tan (2 x)

Use the basic trigonometric identity:

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

= - 1 + \frac{1}{cos ( 2 x )} + tan (2 x)

Use the basic trigonometric identity:

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

= - 1 + \frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )}

Simplify

- 1 + \frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )} : \frac{- cos ( 2 x ) + 1 + sin ( 2 x )}{cos ( 2 x )}

- 1 + \frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )}

Combine the fractions

\frac{1}{cos ( 2 x )} + \frac{sin ( 2 x )}{cos ( 2 x )} : \frac{1 + sin ( 2 x )}{cos ( 2 x )}

Apply rule

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

= \frac{1 + sin ( 2 x )}{cos ( 2 x )}

= - 1 + \frac{sin ( 2 x ) + 1}{cos ( 2 x )}

Convert element to fraction:

1 = \frac{1 cos ( 2 x )}{cos ( 2 x )}

= - \frac{1 \cdot cos ( 2 x )}{cos ( 2 x )} + \frac{1 + sin ( 2 x )}{cos ( 2 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 ( 2 x ) + 1 + sin ( 2 x )}{cos ( 2 x )}

Multiply:

1 \cdot cos (2 x) = cos (2 x)

= \frac{- cos ( 2 x ) + 1 + sin ( 2 x )}{cos ( 2 x )}

= \frac{- cos ( 2 x ) + 1 + sin ( 2 x )}{cos ( 2 x )}

\frac{1 - cos ( 2 x ) + sin ( 2 x )}{cos ( 2 x )} = 0

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

1 - cos (2 x) + sin (2 x) = 0

Rewrite using trig identities

1 - cos (2 x) + sin (2 x)

sin (2 x) - cos (2 x) = 2 sin (2 x - \frac{π}{4})

sin (2 x) - cos (2 x)

Rewrite as

= 2 (\frac{1}{2} sin (2 x) - \frac{1}{2} cos (2 x))

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (2 x) - sin (\frac{π}{4}) cos (2 x))

Use the Angle Sum identity:

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

= 2 sin (2 x - \frac{π}{4})

= 1 + 2 sin (2 x - \frac{π}{4})

1 + 2 sin (2 x - \frac{π}{4}) = 0

Move

1

to the right side

1 + 2 sin (2 x - \frac{π}{4}) = 0

Subtract

1

from both sides

1 + 2 sin (2 x - \frac{π}{4}) - 1 = 0 - 1

Simplify

2 sin (2 x - \frac{π}{4}) = - 1

2 sin (2 x - \frac{π}{4}) = - 1

Divide both sides by

2

2 sin (2 x - \frac{π}{4}) = - 1

Divide both sides by

2

\frac{2 sin ( 2 x - \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( 2 x - \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplify

\frac{2 sin ( 2 x - \frac{π}{4} )}{2} : sin (2 x - \frac{π}{4})

\frac{2 sin ( 2 x - \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (2 x - \frac{π}{4})

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (2 x - \frac{π}{4}) = - \frac{2}{2}

sin (2 x - \frac{π}{4}) = - \frac{2}{2}

sin (2 x - \frac{π}{4}) = - \frac{2}{2}

General solutions for

sin (2 x - \frac{π}{4}) = - \frac{2}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x - \frac{π}{4} = \frac{5 π}{4} + 2 πn, 2 x - \frac{π}{4} = \frac{7 π}{4} + 2 πn

2 x - \frac{π}{4} = \frac{5 π}{4} + 2 πn, 2 x - \frac{π}{4} = \frac{7 π}{4} + 2 πn

Solve

2 x - \frac{π}{4} = \frac{5 π}{4} + 2 πn : x = πn + \frac{3 π}{4}

2 x - \frac{π}{4} = \frac{5 π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

2 x - \frac{π}{4} = \frac{5 π}{4} + 2 πn

Add

\frac{π}{4}

to both sides

2 x - \frac{π}{4} + \frac{π}{4} = \frac{5 π}{4} + 2 πn + \frac{π}{4}

Simplify

2 x - \frac{π}{4} + \frac{π}{4} = \frac{5 π}{4} + 2 πn + \frac{π}{4}

Simplify

2 x - \frac{π}{4} + \frac{π}{4} : 2 x

2 x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= 2 x

Simplify

\frac{5 π}{4} + 2 πn + \frac{π}{4} : 2 πn + \frac{3 π}{2}

\frac{5 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{5 π}{4}

Combine the fractions

\frac{π}{4} + \frac{5 π}{4} : \frac{3 π}{2}

Apply rule

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

= \frac{π + 5 π}{4}

Add similar elements:

π + 5 π = 6 π

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

2 x = 2 πn + \frac{3 π}{2}

2 x = 2 πn + \frac{3 π}{2}

2 x = 2 πn + \frac{3 π}{2}

Divide both sides by

2

2 x = 2 πn + \frac{3 π}{2}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

= πn + \frac{3 π}{4}

x = πn + \frac{3 π}{4}

x = πn + \frac{3 π}{4}

x = πn + \frac{3 π}{4}

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

2 x - \frac{π}{4} + \frac{π}{4} : 2 x

2 x - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= 2 x

Simplify

\frac{7 π}{4} + 2 πn + \frac{π}{4} : 2 πn + 2 π

\frac{7 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{7 π}{4}

Combine the fractions

\frac{π}{4} + \frac{7 π}{4} : 2 π

Apply rule

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

= \frac{π + 7 π}{4}

Add similar elements:

π + 7 π = 8 π

= \frac{8 π}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2 π

= 2 πn + 2 π

2 x = 2 πn + 2 π

2 x = 2 πn + 2 π

2 x = 2 πn + 2 π

Divide both sides by

2

2 x = 2 πn + 2 π

Divide both sides by

2

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

Simplify

x = πn + π

x = πn + π

x = πn + \frac{3 π}{4}, x = πn + π

Since the equation is undefined for:

πn + \frac{3 π}{4}

x = πn + π

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Frequently Asked Questions (FAQ)

What is the general solution for tan(2x)+sec(2x)=1 ?
The general solution for tan(2x)+sec(2x)=1 is x=pin+pi