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

The general solution for 2cot(x)=csc^2(x) is x= pi/4+pin

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

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

csc^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

csc^{2} (x) = 1 + cot^{2} (x)

= - (1 + cot^{2} (x)) + 2 cot (x)

- (1 + cot^{2} (x)) : - 1 - cot^{2} (x)

- (1 + cot^{2} (x))

Distribute parentheses

= - (1) - (cot^{2} (x))

Apply minus-plus rules

+ (- a) = - a

= - 1 - cot^{2} (x)

= - 1 - cot^{2} (x) + 2 cot (x)

- 1 - cot^{2} (x) + 2 cot (x) = 0

Solve by substitution

- 1 - cot^{2} (x) + 2 cot (x) = 0

Let:

cot (x) = u

- 1 - u^{2} + 2 u = 0

- 1 - u^{2} + 2 u = 0 : u = 1

- 1 - u^{2} + 2 u = 0

Write in the standard form

a x^{2} + b x + c = 0

- u^{2} + 2 u - 1 = 0

Solve with the quadratic formula

- u^{2} + 2 u - 1 = 0

Quadratic Equation Formula:

For

a = - 1, b = 2, c = - 1

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 1 ) ( - 1 )}{2 ( - 1 )}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 1 ) ( - 1 )}{2 ( - 1 )}

2^{2} - 4 (- 1) (- 1) = 0

2^{2} - 4 (- 1) (- 1)

Apply rule

- (- a) = a

= 2^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2^{2} - 4

2^{2} = 4

= 4 - 4

Subtract the numbers:

4 - 4 = 0

= 0

u_{1, 2} = \frac{- 2 \pm 0}{2 ( - 1 )}

u = \frac{- 2}{2 ( - 1 )}

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

\frac{- 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 2}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{- 2}

Apply the fraction rule:

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

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = 1

The solution to the quadratic equation is:

u = 1

Substitute back

u = cot (x)

cot (x) = 1

cot (x) = 1

cot (x) = 1 : x = \frac{π}{4} + πn

cot (x) = 1

General solutions for

cot (x) = 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

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn

Popular Examples

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2sin(x)(sin(x)-1)=3cos^2(x)

2 sin (x) (sin (x) - 1) = 3 cos^{2} (x)

2sin^2(x)=2sin(x)+4

2 sin^{2} (x) = 2 sin (x) + 4

Frequently Asked Questions (FAQ)

What is the general solution for 2cot(x)=csc^2(x) ?
The general solution for 2cot(x)=csc^2(x) is x= pi/4+pin