What is the general solution for cot^2(x)=(1+sin(x))/(1+cos(x)) ?

The general solution for cot^2(x)=(1+sin(x))/(1+cos(x)) is x=(3pi)/2+2pin,x= pi/4+pin

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cot^2(x)=(1+sin(x))/(1+cos(x))

Solution

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

Solution

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

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

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

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Expand

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = cot^{2} (x), b = 1, c = cos (x)

= cot^{2} (x) \cdot 1 + cot^{2} (x) cos (x)

= 1 \cdot cot^{2} (x) + cot^{2} (x) cos (x)

Multiply:

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

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

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

- (1 + sin (x)) : - 1 - sin (x)

- (1 + sin (x))

Distribute parentheses

= - (1) - (sin (x))

Apply minus-plus rules

+ (- a) = - a

= - 1 - sin (x)

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

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

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

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

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

- 1 + (\frac{cos ( x )}{sin ( x )})^{2} - sin (x) + cos (x) (\frac{cos ( x )}{sin ( x )})^{2} : \frac{- sin ^{2} ( x ) + cos ^{2} ( x ) + cos ^{3} ( x ) - sin ^{3} ( x )}{sin ^{2} ( x )}

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

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

cos (x) (\frac{cos ( x )}{sin ( x )})^{2} = \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

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

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

Multiply fractions:

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

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

cos^{2} (x) cos (x) = cos^{3} (x)

cos^{2} (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos (x) = cos^{2 + 1} (x)

= cos^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= cos^{3} (x)

= \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

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

Combine the fractions

\frac{cos ^{2} ( x )}{sin ^{2} ( x )} + \frac{cos ^{3} ( x )}{sin ^{2} ( x )} : \frac{cos ^{2} ( x ) + cos ^{3} ( x )}{sin ^{2} ( x )}

Apply rule

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

= \frac{cos ^{2} ( x ) + cos ^{3} ( x )}{sin ^{2} ( x )}

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

Convert element to fraction:

1 = \frac{1 sin ^{2} ( x )}{sin ^{2} ( x )}, sin (x) = \frac{sin ( x ) sin ^{2} ( x )}{sin ^{2} ( x )}

= - \frac{1 \cdot sin ^{2} ( x )}{sin ^{2} ( x )} + \frac{cos ^{2} ( x ) + cos ^{3} ( x )}{sin ^{2} ( x )} - \frac{sin ( x ) sin ^{2} ( x )}{sin ^{2} ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot sin ^{2} ( x ) + cos ^{2} ( x ) + cos ^{3} ( x ) - sin ( x ) sin ^{2} ( x )}{sin ^{2} ( x )}

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

- 1 \cdot sin^{2} (x) + cos^{2} (x) + cos^{3} (x) - sin (x) sin^{2} (x)

1 \cdot sin^{2} (x) = sin^{2} (x)

1 \cdot sin^{2} (x)

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

= sin^{2} (x)

sin (x) sin^{2} (x) = sin^{3} (x)

sin (x) sin^{2} (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin^{2} (x) = sin^{1 + 2} (x)

= sin^{1 + 2} (x)

Add the numbers:

1 + 2 = 3

= sin^{3} (x)

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

= \frac{- sin ^{2} ( x ) + cos ^{2} ( x ) + cos ^{3} ( x ) - sin ^{3} ( x )}{sin ^{2} ( x )}

= \frac{- sin ^{2} ( x ) + cos ^{2} ( x ) + cos ^{3} ( x ) - sin ^{3} ( x )}{sin ^{2} ( x )}

\frac{cos ^{2} ( x ) + cos ^{3} ( x ) - sin ^{2} ( x ) - sin ^{3} ( x )}{sin ^{2} ( x )} = 0

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

cos^{2} (x) + cos^{3} (x) - sin^{2} (x) - sin^{3} (x) = 0

Factor

cos^{2} (x) + cos^{3} (x) - sin^{2} (x) - sin^{3} (x) : (- sin (x) + cos (x)) (sin^{2} (x) + sin (x) cos (x) + sin (x) + cos^{2} (x) + cos (x))

cos^{2} (x) + cos^{3} (x) - sin^{2} (x) - sin^{3} (x)

Factor

cos^{3} (x) - sin^{3} (x) : (cos (x) - sin (x)) (cos^{2} (x) + sin (x) cos (x) + sin^{2} (x))

cos^{3} (x) - sin^{3} (x)

Apply Difference of Cubes Formula:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

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

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

Factor

cos^{2} (x) - sin^{2} (x) : (cos (x) + sin (x)) (cos (x) - sin (x))

cos^{2} (x) - sin^{2} (x)

Apply Difference of Two Squares Formula:

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

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

= (cos (x) + sin (x)) (cos (x) - sin (x))

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

Factor out common term

(cos (x) - sin (x))

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

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

Rewrite using trig identities

(- sin (x) + cos (x)) (sin^{2} (x) + sin (x) cos (x) + sin (x) + cos^{2} (x) + cos (x))

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

= (cos (x) + sin (x) + cos (x) sin (x) + 1) (cos (x) - sin (x))

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

Solving each part separately

cos (x) + sin (x) + cos (x) sin (x) + 1 = 0 or cos (x) - sin (x) = 0

cos (x) + sin (x) + cos (x) sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn, x = π + 2 πn

cos (x) + sin (x) + cos (x) sin (x) + 1 = 0

Factor

cos (x) + sin (x) + cos (x) sin (x) + 1 : (1 + sin (x)) (cos (x) + 1)

cos (x) + sin (x) + cos (x) sin (x) + 1

Factor out common term

cos (x)

= cos (x) (1 + sin (x)) + sin (x) + 1

Rewrite as

= (1 + sin (x)) cos (x) + 1 \cdot (1 + sin (x))

Factor out common term

(1 + sin (x))

= (1 + sin (x)) (cos (x) + 1)

(1 + sin (x)) (cos (x) + 1) = 0

Solving each part separately

1 + sin (x) = 0 or cos (x) + 1 = 0

1 + sin (x) = 0 : x = \frac{3 π}{2} + 2 πn

1 + sin (x) = 0

Move

1

to the right side

1 + sin (x) = 0

Subtract

1

from both sides

1 + sin (x) - 1 = 0 - 1

Simplify

sin (x) = - 1

sin (x) = - 1

General solutions for

sin (x) = - 1

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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Move

1

to the right side

cos (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

cos (x) = - 1

cos (x) = - 1

General solutions for

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

Combine all the solutions

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

cos (x) - sin (x) = 0 : x = \frac{π}{4} + πn

cos (x) - sin (x) = 0

Rewrite using trig identities

cos (x) - sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 - tan (x) = 0

1 - tan (x) = 0

Move

1

to the right side

1 - tan (x) = 0

Subtract

1

from both sides

1 - tan (x) - 1 = 0 - 1

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

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

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

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

Since the equation is undefined for:

π + 2 πn

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cot^2(x)=(1+sin(x))/(1+cos(x)) ?
The general solution for cot^2(x)=(1+sin(x))/(1+cos(x)) is x=(3pi)/2+2pin,x= pi/4+pin