English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

csc^2(x)=2*cot^2(x)

Solution

csc^{2} (x) = 2 \cdot cot^{2} (x)

Solution

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

csc^{2} (x) = 2 cot^{2} (x)

Subtract

2 cot^{2} (x)

from both sides

csc^{2} (x) - 2 cot^{2} (x) = 0

Factor

csc^{2} (x) - 2 cot^{2} (x) : (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

csc^{2} (x) - 2 cot^{2} (x)

Rewrite

csc^{2} (x) - 2 cot^{2} (x)

csc^{2} (x) - (2 cot (x))^{2}

csc^{2} (x) - 2 cot^{2} (x)

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= csc^{2} (x) - (2)^{2} cot^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} cot^{2} (x) = (2 cot (x))^{2}

= csc^{2} (x) - (2 cot (x))^{2}

= csc^{2} (x) - (2 cot (x))^{2}

Apply Difference of Two Squares Formula:

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

csc^{2} (x) - (2 cot (x))^{2} = (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

= (csc (x) + 2 cot (x)) (csc (x) - 2 cot (x))

(csc (x) + 2 cot (x)) (csc (x) - 2 cot (x)) = 0

Solving each part separately

csc (x) + 2 cot (x) = 0 or csc (x) - 2 cot (x) = 0

csc (x) + 2 cot (x) = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

csc (x) + 2 cot (x) = 0

Express with sin, cos

csc (x) + cot (x) 2

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

\frac{cos ( x )}{sin ( x )} 2

Multiply fractions:

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

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

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

Apply rule

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

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

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

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

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

1 + cos (x) 2 = 0

Move

1

to the right side

1 + cos (x) 2 = 0

Subtract

1

from both sides

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

Simplify

cos (x) 2 = - 1

cos (x) 2 = - 1

Divide both sides by

2

cos (x) 2 = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{cos ( x ) 2}{2} : cos (x)

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

Cancel the common factor:

2

= cos (x)

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}

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

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

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

General solutions for

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

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

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

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

csc (x) - 2 cot (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

csc (x) - 2 cot (x) = 0

Express with sin, cos

csc (x) - cot (x) 2

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

\frac{cos ( x )}{sin ( x )} 2

Multiply fractions:

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

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

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

Apply rule

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

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

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

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

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

1 - cos (x) 2 = 0

Move

1

to the right side

1 - cos (x) 2 = 0

Subtract

1

from both sides

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

Simplify

- cos (x) 2 = - 1

- cos (x) 2 = - 1

Divide both sides by

- 2

- cos (x) 2 = - 1

Divide both sides by

- 2

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

Simplify

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

Simplify

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

\frac{- cos ( x ) 2}{- 2}

Apply the fraction rule:

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

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

Cancel the common factor:

2

= cos (x)

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}

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

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

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

General solutions for

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

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

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

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

Combine all the solutions

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Popular Examples

3sin(2x)=cos(2x)

3 sin (2 x) = cos (2 x)

3*2=(50sin(x))/(9.8)

3 \cdot 2 = \frac{50 sin ( x )}{9.8}

cos(2x+5)=sin(2x)

cos (2 x + 5) = sin (2 x)

sin(2θ)+3cos(θ)=cos(θ)

sin (2 θ) + 3 cos (θ) = cos (θ)

1+cos(θ)=1+sin(θ)

1 + cos (θ) = 1 + sin (θ)