What is the general solution for (1-csc(θ))/(cot(θ))=tan(θ)-csc(θ) ?

The general solution for (1-csc(θ))/(cot(θ))=tan(θ)-csc(θ) is θ= pi/4+pin

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(1-csc(θ))/(cot(θ))=tan(θ)-csc(θ)

Solution

\frac{1 - csc ( θ )}{cot ( θ )} = tan (θ) - csc (θ)

Solution

θ = \frac{π}{4} + πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

\frac{1 - csc ( θ )}{cot ( θ )} = tan (θ) - csc (θ)

Subtract

tan (θ) - csc (θ)

from both sides

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

Simplify

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

\frac{1 - csc ( θ )}{cot ( θ )} - tan (θ) + csc (θ)

Convert element to fraction:

tan (θ) = \frac{tan ( θ ) cot ( θ )}{cot ( θ )}, csc (θ) = \frac{csc ( θ ) cot ( θ )}{cot ( θ )}

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

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

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

1 - csc (θ) - tan (θ) cot (θ) + csc (θ) cot (θ) = 0

Rewrite using trig identities

1 - csc (θ) + cot (θ) csc (θ) - cot (θ) tan (θ)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

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

Simplify

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

1 - csc (θ) + cot (θ) csc (θ) - cot (θ) \frac{1}{cot ( θ )}

cot (θ) \frac{1}{cot ( θ )} = 1

cot (θ) \frac{1}{cot ( θ )}

Multiply fractions:

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

= \frac{1 \cdot cot ( θ )}{cot ( θ )}

Cancel the common factor:

cot (θ)

= 1

= 1 - csc (θ) + cot (θ) csc (θ) - 1

Group like terms

= - csc (θ) + cot (θ) csc (θ) + 1 - 1

1 - 1 = 0

= cot (θ) csc (θ) - csc (θ)

= cot (θ) csc (θ) - csc (θ)

- csc (θ) + cot (θ) csc (θ) = 0

Factor

- csc (θ) + cot (θ) csc (θ) : csc (θ) (cot (θ) - 1)

- csc (θ) + cot (θ) csc (θ)

Factor out common term

csc (θ)

= csc (θ) (- 1 + cot (θ))

csc (θ) (cot (θ) - 1) = 0

Solving each part separately

csc (θ) = 0 or cot (θ) - 1 = 0

csc (θ) = 0 :

No Solution

csc (θ) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

cot (θ) - 1 = 0 : θ = \frac{π}{4} + πn

cot (θ) - 1 = 0

Move

1

to the right side

cot (θ) - 1 = 0

Add

1

to both sides

cot (θ) - 1 + 1 = 0 + 1

Simplify

cot (θ) = 1

cot (θ) = 1

General solutions for

cot (θ) = 1

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

Combine all the solutions

θ = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (1-csc(θ))/(cot(θ))=tan(θ)-csc(θ) ?
The general solution for (1-csc(θ))/(cot(θ))=tan(θ)-csc(θ) is θ= pi/4+pin