What is the general solution for 2cot(x)=csc(x) ?

The general solution for 2cot(x)=csc(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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2cot(x)=csc(x)

2 cot (x) = csc (x)

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 cot (x) = csc (x)

Subtract

csc (x)

from both sides

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

Express with sin, cos

- csc (x) + 2 cot (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

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

- 1 + 2 cos (x) = 0

Move

1

to the right side

- 1 + 2 cos (x) = 0

Add

1

to both sides

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

Simplify

2 cos (x) = 1

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

cos (x) = \frac{1}{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}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

- 2 sec^{2} (x) tan (x) = 0

5 tan^{2} (θ) - tan (θ) + 12 = 8 tan (θ) + 8

16 sin^{2} (θ) - 4 = 0

22 cos (x) - 2 = 0

cos^{2} (x) + 2 sin (x) = 0

What is the general solution for 2cot(x)=csc(x) ?
The general solution for 2cot(x)=csc(x) is x= pi/3+2pin,x=(5pi)/3+2pin