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Popular >

(cos^2(x))/(1-cos(x))=1+cos(x)

Solution

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

Solution

x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

Degrees

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

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

\frac{u ^{2}}{1 - u} = 1 + u

\frac{u ^{2}}{1 - u} = 1 + u : u = \frac{1}{2}, u = - \frac{1}{2}

\frac{u ^{2}}{1 - u} = 1 + u

Multiply both sides by

1 - u

\frac{u ^{2}}{1 - u} = 1 + u

Multiply both sides by

1 - u

\frac{u ^{2}}{1 - u} (1 - u) = 1 \cdot (1 - u) + u (1 - u)

Simplify

\frac{u ^{2}}{1 - u} (1 - u) = 1 \cdot (1 - u) + u (1 - u)

Simplify

\frac{u ^{2}}{1 - u} (1 - u) : u^{2}

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

Multiply fractions:

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

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

Cancel the common factor:

1 - u

= u^{2}

Simplify

1 \cdot (1 - u) : 1 - u

1 \cdot (1 - u)

Multiply:

1 \cdot (1 - u) = (1 - u)

= (1 - u)

Remove parentheses:

(a) = a

= 1 - u

u^{2} = 1 - u + u (1 - u)

u^{2} = 1 - u + u (1 - u)

u^{2} = 1 - u + u (1 - u)

Solve

u^{2} = 1 - u + u (1 - u) : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = 1 - u + u (1 - u)

Expand

1 - u + u (1 - u) : 1 - u^{2}

1 - u + u (1 - u)

Expand

u (1 - u) : u - u^{2}

u (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = u, b = 1, c = u

= u \cdot 1 - uu

= 1 \cdot u - uu

Simplify

1 \cdot u - uu : u - u^{2}

1 \cdot u - uu

1 \cdot u = u

1 \cdot u

Multiply:

1 \cdot u = u

= u

uu = u^{2}

uu

Apply exponent rule:

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

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

= u - u^{2}

= u - u^{2}

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

Add similar elements:

- u + u = 0

= 1 - u^{2}

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

Move

u^{2}

to the left side

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

Add

u^{2}

to both sides

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

Simplify

2 u^{2} = 1

2 u^{2} = 1

Divide both sides by

2

2 u^{2} = 1

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{1}{2}

Simplify

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{2}, u = - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 1

Take the denominator(s) of

\frac{u ^{2}}{1 - u}

and compare to zero

Solve

1 - u = 0 : u = 1

1 - u = 0

Move

1

to the right side

1 - u = 0

Subtract

1

from both sides

1 - u - 1 = 0 - 1

Simplify

- u = - 1

- u = - 1

Divide both sides by

- 1

- u = - 1

Divide both sides by

- 1

\frac{- u}{- 1} = \frac{- 1}{- 1}

Simplify

u = 1

u = 1

The following points are undefined

u = 1

Combine undefined points with solutions:

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = cos (x)

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

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

cos (x) = \frac{1}{2} : x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn

cos (x) = - \frac{1}{2} : x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

Combine all the solutions

x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn, x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

Show solutions in decimal form

x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

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