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

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

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1-tan(x)=sec(x)

Solution

1 - tan (x) = sec (x)

Solution

x = 2 πn, x = 2 π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

1 - tan (x) = sec (x)

Subtract

sec (x)

from both sides

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

Express with sin, cos

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

Simplify

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

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

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

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

Add

sin (x)

to both sides

cos (x) - 1 = sin (x)

Square both sides

(cos (x) - 1)^{2} = sin^{2} (x)

Subtract

sin^{2} (x)

from both sides

(cos (x) - 1)^{2} - sin^{2} (x) = 0

Factor

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

(cos (x) - 1)^{2} - sin^{2} (x)

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

Refine

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

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

Solving each part separately

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

cos (x) - 1 + sin (x) = 0 : x = 2 πn, x = 2 πn + \frac{π}{2}

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

Rewrite using trig identities

cos (x) - 1 + sin (x)

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

sin (x) + cos (x)

Rewrite as

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

Use the Angle Sum identity:

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

= 2 sin (x + \frac{π}{4})

= - 1 + 2 sin (x + \frac{π}{4})

- 1 + 2 sin (x + \frac{π}{4}) = 0

Move

1

to the right side

- 1 + 2 sin (x + \frac{π}{4}) = 0

Add

1

to both sides

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

Simplify

2 sin (x + \frac{π}{4}) = 1

2 sin (x + \frac{π}{4}) = 1

Divide both sides by

2

2 sin (x + \frac{π}{4}) = 1

Divide both sides by

2

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

Simplify

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

Simplify

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

\frac{2 sin ( x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x + \frac{π}{4})

Simplify

\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}

sin (x + \frac{π}{4}) = \frac{2}{2}

sin (x + \frac{π}{4}) = \frac{2}{2}

sin (x + \frac{π}{4}) = \frac{2}{2}

General solutions for

sin (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}

x + \frac{π}{4} = \frac{π}{4} + 2 πn, x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

x + \frac{π}{4} = \frac{π}{4} + 2 πn, x + \frac{π}{4} = \frac{3 π}{4} + 2 πn

Solve

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

x + \frac{π}{4} = \frac{π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

x = 2 πn

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

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

\frac{3 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{3 π}{4}

Combine the fractions

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

Apply rule

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

= \frac{- π + 3 π}{4}

Add similar elements:

- π + 3 π = 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn, x = 2 πn + \frac{π}{2}

cos (x) - 1 - sin (x) = 0 : x = \frac{3 π}{2} + 2 πn, x = 2 π + 2 πn

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

Rewrite using trig identities

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

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

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

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

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

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

2 sin (\frac{x + \frac{π}{2} - x}{2}) sin (\frac{x - ( \frac{π}{2} - x )}{2})

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

\frac{x + \frac{π}{2} - x}{2}

x + \frac{π}{2} - x = \frac{π}{2}

x + \frac{π}{2} - x

Group like terms

= x - x + \frac{π}{2}

Add similar elements:

x - x = 0

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

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

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

\frac{x - ( \frac{π}{2} - x )}{2}

Expand

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

x - (\frac{π}{2} - x)

- (\frac{π}{2} - x) : - \frac{π}{2} + x

- (\frac{π}{2} - x)

Distribute parentheses

= - (\frac{π}{2}) - (- x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + x

= x - \frac{π}{2} + x

Simplify

x - \frac{π}{2} + x : 2 x - \frac{π}{2}

x - \frac{π}{2} + x

Group like terms

= x + x - \frac{π}{2}

Add similar elements:

x + x = 2 x

= 2 x - \frac{π}{2}

= 2 x - \frac{π}{2}

= \frac{2 x - \frac{π}{2}}{2}

Join

2 x - \frac{π}{2} : \frac{4 x - π}{2}

2 x - \frac{π}{2}

Convert element to fraction:

2 x = \frac{2 x 2}{2}

= \frac{2 x \cdot 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 x \cdot 2 - π}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π}{2}

= \frac{\frac{4 x - π}{2}}{2}

Apply the fraction rule:

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

= \frac{4 x - π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π}{4}

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

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\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}

= \frac{2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 sin (\frac{4 x - π}{4})

= - 1 - 2 sin (\frac{4 x - π}{4})

- 1 - 2 sin (\frac{4 x - π}{4}) = 0

Move

1

to the right side

- 1 - 2 sin (\frac{4 x - π}{4}) = 0

Add

1

to both sides

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

Simplify

- 2 sin (\frac{4 x - π}{4}) = 1

- 2 sin (\frac{4 x - π}{4}) = 1

Divide both sides by

- 2

- 2 sin (\frac{4 x - π}{4}) = 1

Divide both sides by

- 2

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

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

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

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

General solutions for

sin (\frac{4 x - 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}

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

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

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( 4 x - π )}{4} : 4 x - π

\frac{4 ( 4 x - π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 4 x - π

Simplify

4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn : 5 π + 8 πn

4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{4} = 5 π

4 \cdot \frac{5 π}{4}

Multiply fractions:

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

= \frac{5 π 4}{4}

Cancel the common factor:

4

= 5 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 5 π + 8 πn

4 x - π = 5 π + 8 πn

4 x - π = 5 π + 8 πn

4 x - π = 5 π + 8 πn

Move

π

to the right side

4 x - π = 5 π + 8 πn

Add

π

to both sides

4 x - π + π = 5 π + 8 πn + π

Simplify

4 x = 6 π + 8 πn

4 x = 6 π + 8 πn

Divide both sides by

4

4 x = 6 π + 8 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{6 π}{4} + \frac{8 πn}{4}

Simplify

\frac{4 x}{4} = \frac{6 π}{4} + \frac{8 πn}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{6 π}{4} + \frac{8 πn}{4} : \frac{3 π}{2} + 2 πn

\frac{6 π}{4} + \frac{8 πn}{4}

Cancel

\frac{6 π}{4} : \frac{3 π}{2}

\frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + \frac{8 πn}{4}

Divide the numbers:

\frac{8}{4} = 2

= \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

\frac{4 ( 4 x - π )}{4} : 4 x - π

\frac{4 ( 4 x - π )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 4 x - π

Simplify

4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn : 7 π + 8 πn

4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn

4 \cdot \frac{7 π}{4} = 7 π

4 \cdot \frac{7 π}{4}

Multiply fractions:

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

= \frac{7 π 4}{4}

Cancel the common factor:

4

= 7 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 7 π + 8 πn

4 x - π = 7 π + 8 πn

4 x - π = 7 π + 8 πn

4 x - π = 7 π + 8 πn

Move

π

to the right side

4 x - π = 7 π + 8 πn

Add

π

to both sides

4 x - π + π = 7 π + 8 πn + π

Simplify

4 x = 8 π + 8 πn

4 x = 8 π + 8 πn

Divide both sides by

4

4 x = 8 π + 8 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{8 π}{4} + \frac{8 πn}{4}

Simplify

x = 2 π + 2 πn

x = 2 π + 2 πn

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

Combine all the solutions

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

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

1 - tan (x) = sec (x)

Remove the ones that don't agree with the equation.

Check the solution

2 πn :

True

2 πn

Plug in

n = 1

2 π 1

For

1 - tan (x) = sec (x)

plug in

x = 2 π 1

1 - tan (2 π 1) = sec (2 π 1)

Refine

1 = 1

\Rightarrow True

Check the solution

2 πn + \frac{π}{2} :

False

2 πn + \frac{π}{2}

Plug in

n = 1

2 π 1 + \frac{π}{2}

For

1 - tan (x) = sec (x)

plug in

x = 2 π 1 + \frac{π}{2}

1 - tan (2 π 1 + \frac{π}{2}) = sec (2 π 1 + \frac{π}{2})

Refine

- \infty = \infty

\Rightarrow False

Check the solution

\frac{3 π}{2} + 2 πn :

True

\frac{3 π}{2} + 2 πn

Plug in

n = 1

\frac{3 π}{2} + 2 π 1

For

1 - tan (x) = sec (x)

plug in

x = \frac{3 π}{2} + 2 π 1

1 - tan (\frac{3 π}{2} + 2 π 1) = sec (\frac{3 π}{2} + 2 π 1)

Refine

- \infty = - \infty

\Rightarrow True

Check the solution

2 π + 2 πn :

True

2 π + 2 πn

Plug in

n = 1

2 π + 2 π 1

For

1 - tan (x) = sec (x)

plug in

x = 2 π + 2 π 1

1 - tan (2 π + 2 π 1) = sec (2 π + 2 π 1)

Refine

1 = 1

\Rightarrow True

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

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

x = 2 πn, x = 2 π + 2 πn

Popular Examples

sec((3θ)/2)=-2,0<= θ<2pi

sec (\frac{3 θ}{2}) = - 2, 0 \leq θ < 2 π

tan(x-pi/6)=sqrt(3)

tan (x - \frac{π}{6}) = 3

-5cos(5x)=0

- 5 cos (5 x) = 0

2sin(x)tan(x)+5=4sec(x)

2 sin (x) tan (x) + 5 = 4 sec (x)

cos^2(x)-cos(x)-2=0,0<= x<= 2pi

cos^{2} (x) - cos (x) - 2 = 0, 0 \leq x \leq 2 π

Frequently Asked Questions (FAQ)

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