What is the general solution for sec(θ)+1/(cot(θ))=1 ?

The general solution for sec(θ)+1/(cot(θ))=1 is No Solution for θ\in\mathbb{R}

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sec(θ)+1/(cot(θ))=1

Solution

sec (θ) + \frac{1}{cot ( θ )} = 1

Solution

No Solution for θ \in R

Solution steps

sec (θ) + \frac{1}{cot ( θ )} = 1

Subtract

1

from both sides

sec (θ) + \frac{1}{cot ( θ )} - 1 = 0

Simplify

sec (θ) + \frac{1}{cot ( θ )} - 1 : \frac{sec ( θ ) cot ( θ ) + 1 - cot ( θ )}{cot ( θ )}

sec (θ) + \frac{1}{cot ( θ )} - 1

Convert element to fraction:

sec (θ) = \frac{sec ( θ ) cot ( θ )}{cot ( θ )}, 1 = \frac{1 cot ( θ )}{cot ( θ )}

= \frac{sec ( θ ) cot ( θ )}{cot ( θ )} + \frac{1}{cot ( θ )} - \frac{1 \cdot cot ( θ )}{cot ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{sec ( θ ) cot ( θ ) + 1 - 1 \cdot cot ( θ )}{cot ( θ )}

Multiply:

1 \cdot cot (θ) = cot (θ)

= \frac{sec ( θ ) cot ( θ ) + 1 - cot ( θ )}{cot ( θ )}

\frac{sec ( θ ) cot ( θ ) + 1 - cot ( θ )}{cot ( θ )} = 0

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

sec (θ) cot (θ) + 1 - cot (θ) = 0

Express with sin, cos

1 - cot (θ) + cot (θ) sec (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

\frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{cos ( θ )} = \frac{1}{sin ( θ )}

\frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{cos ( θ )}

Multiply fractions:

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

= \frac{cos ( θ ) \cdot 1}{sin ( θ ) cos ( θ )}

Cancel the common factor:

cos (θ)

= \frac{1}{sin ( θ )}

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

Combine the fractions

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

Apply rule

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

= \frac{- cos ( θ ) + 1}{sin ( θ )}

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

Convert element to fraction:

1 = \frac{1 sin ( θ )}{sin ( θ )}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sin (θ) = sin (θ)

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

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

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

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

1 - cos (θ) + sin (θ) = 0

Rewrite using trig identities

1 - cos (θ) + sin (θ)

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

sin (θ) - cos (θ)

Rewrite as

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

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 (θ) - sin (\frac{π}{4}) cos (θ))

Use the Angle Sum identity:

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

= 2 sin (θ - \frac{π}{4})

= 1 + 2 sin (θ - \frac{π}{4})

1 + 2 sin (θ - \frac{π}{4}) = 0

Move

1

to the right side

1 + 2 sin (θ - \frac{π}{4}) = 0

Subtract

1

from both sides

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

Simplify

2 sin (θ - \frac{π}{4}) = - 1

2 sin (θ - \frac{π}{4}) = - 1

Divide both sides by

2

2 sin (θ - \frac{π}{4}) = - 1

Divide both sides by

2

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

Simplify

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

Simplify

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

\frac{2 sin ( θ - \frac{π}{4} )}{2}

Cancel the common factor:

2

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

sin (θ - \frac{π}{4}) = - \frac{2}{2}

sin (θ - \frac{π}{4}) = - \frac{2}{2}

General solutions for

sin (θ - \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}

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

θ - \frac{π}{4} + \frac{π}{4} : θ

θ - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= θ

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}

θ = 2 πn + \frac{3 π}{2}

θ = 2 πn + \frac{3 π}{2}

θ = 2 πn + \frac{3 π}{2}

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

θ - \frac{π}{4} + \frac{π}{4} : θ

θ - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= θ

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

θ = 2 πn + 2 π

θ = 2 πn + 2 π

θ = 2 πn + 2 π

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

Since the equation is undefined for:

2 πn + \frac{3 π}{2}, 2 πn + 2 π

No Solution for θ \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sec(θ)+1/(cot(θ))=1 ?
The general solution for sec(θ)+1/(cot(θ))=1 is No Solution for θ\in\mathbb{R}