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1/(1+cos(x))+1/(1-cos(x))=32

Solution

\frac{1}{1 + cos ( x )} + \frac{1}{1 - cos ( x )} = 32

Solution

x = 2.88891 \dots + 2 πn, x = - 2.88891 \dots + 2 πn, x = 0.25268 \dots + 2 πn, x = 2 π - 0.25268 \dots + 2 πn

Degrees

x = 165.52248 \dots^{\circ} + 36 0^{\circ} n, x = - 165.52248 \dots^{\circ} + 36 0^{\circ} n, x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 345.52248 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{1 + cos ( x )} + \frac{1}{1 - cos ( x )} = 32

Solve by substitution

\frac{1}{1 + cos ( x )} + \frac{1}{1 - cos ( x )} = 32

Let:

cos (x) = u

\frac{1}{1 + u} + \frac{1}{1 - u} = 32

\frac{1}{1 + u} + \frac{1}{1 - u} = 32 : u = - \frac{15}{4}, u = \frac{15}{4}

\frac{1}{1 + u} + \frac{1}{1 - u} = 32

Multiply by LCM

\frac{1}{1 + u} + \frac{1}{1 - u} = 32

Find Least Common Multiplier of

1 + u, 1 - u : (u + 1) (- u + 1)

1 + u, 1 - u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

1 + u

1 - u

= (u + 1) (- u + 1)

Multiply by LCM=

(u + 1) (- u + 1)

\frac{1}{1 + u} (u + 1) (- u + 1) + \frac{1}{1 - u} (u + 1) (- u + 1) = 32 (u + 1) (- u + 1)

Simplify

\frac{1}{1 + u} (u + 1) (- u + 1) + \frac{1}{1 - u} (u + 1) (- u + 1) = 32 (u + 1) (- u + 1)

Simplify

\frac{1}{1 + u} (u + 1) (- u + 1) : - u + 1

\frac{1}{1 + u} (u + 1) (- u + 1)

Multiply fractions:

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

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

Cancel the common factor:

u + 1

= 1 \cdot (- u + 1)

Refine

= - u + 1

Simplify

\frac{1}{1 - u} (u + 1) (- u + 1) : u + 1

\frac{1}{1 - u} (u + 1) (- u + 1)

Multiply fractions:

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

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

Cancel the common factor:

- u + 1

= 1 \cdot (u + 1)

Refine

= u + 1

- u + 1 + u + 1 = 32 (u + 1) (- u + 1)

- u + 1 + u + 1 = 2

- u + 1 + u + 1

Group like terms

= - u + u + 1 + 1

Add similar elements:

- u + u = 0

= 1 + 1

Add the numbers:

1 + 1 = 2

= 2

2 = 32 (u + 1) (- u + 1)

2 = 32 (u + 1) (- u + 1)

2 = 32 (u + 1) (- u + 1)

Solve

2 = 32 (u + 1) (- u + 1) : u = - \frac{15}{4}, u = \frac{15}{4}

2 = 32 (u + 1) (- u + 1)

Expand

32 (u + 1) (- u + 1) : 32 - 32 u^{2}

32 (u + 1) (- u + 1)

Expand

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

(u + 1) (- u + 1)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = u

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - u^{2}

= 32 (1 - u^{2})

Expand

32 (1 - u^{2}) : 32 - 32 u^{2}

32 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 32, b = 1, c = u^{2}

= 32 \cdot 1 - 32 u^{2}

Multiply the numbers:

32 \cdot 1 = 32

= 32 - 32 u^{2}

= 32 - 32 u^{2}

2 = 32 - 32 u^{2}

Switch sides

32 - 32 u^{2} = 2

Move

2

to the left side

32 - 32 u^{2} = 2

Subtract

2

from both sides

32 - 32 u^{2} - 2 = 2 - 2

Simplify

- 32 u^{2} + 30 = 0

- 32 u^{2} + 30 = 0

Solve with the quadratic formula

- 32 u^{2} + 30 = 0

Quadratic Equation Formula:

For

a = - 32, b = 0, c = 30

u_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 ( - 32 ) \cdot 30}{2 ( - 32 )}

u_{1, 2} = \frac{- 0 \pm 0 ^{2} - 4 ( - 32 ) \cdot 30}{2 ( - 32 )}

0^{2} - 4 (- 32) \cdot 30 = 1615

0^{2} - 4 (- 32) \cdot 30

Apply rule

0^{a} = 0

0^{2} = 0

= 0 - 4 (- 32) \cdot 30

Apply rule

- (- a) = a

= 0 + 4 \cdot 32 \cdot 30

Multiply the numbers:

4 \cdot 32 \cdot 30 = 3840

= 0 + 3840

Add the numbers:

0 + 3840 = 3840

= 3840

Prime factorization of

3840 : 2^{8} \cdot 3 \cdot 5

3840

3840

divides by

23840 = 1920 \cdot 2

= 2 \cdot 1920

1920

divides by

21920 = 960 \cdot 2

= 2 \cdot 2 \cdot 960

960

divides by

2960 = 480 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 480

480

divides by

2480 = 240 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 240

240

divides by

2240 = 120 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 120

120

divides by

2120 = 60 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 60

60

divides by

260 = 30 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 30

30

divides by

230 = 15 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

= 2^{8} \cdot 3 \cdot 5

= 2^{8} \cdot 3 \cdot 5

Apply radical rule:

= 2^{8} 3 \cdot 5

Apply radical rule:

2^{8} = 2^{\frac{8}{2}} = 2^{4}

= 2^{4} 3 \cdot 5

Refine

= 1615

u_{1, 2} = \frac{- 0 \pm 16 15}{2 ( - 32 )}

Separate the solutions

u_{1} = \frac{- 0 + 16 15}{2 ( - 32 )}, u_{2} = \frac{- 0 - 16 15}{2 ( - 32 )}

u = \frac{- 0 + 16 15}{2 ( - 32 )} : - \frac{15}{4}

\frac{- 0 + 16 15}{2 ( - 32 )}

Remove parentheses:

(- a) = - a

= \frac{- 0 + 16 15}{- 2 \cdot 32}

- 0 + 1615 = 1615

= \frac{16 15}{- 2 \cdot 32}

Multiply the numbers:

2 \cdot 32 = 64

= \frac{16 15}{- 64}

Apply the fraction rule:

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

= - \frac{16 15}{64}

Cancel the common factor:

16

= - \frac{15}{4}

u = \frac{- 0 - 16 15}{2 ( - 32 )} : \frac{15}{4}

\frac{- 0 - 16 15}{2 ( - 32 )}

Remove parentheses:

(- a) = - a

= \frac{- 0 - 16 15}{- 2 \cdot 32}

- 0 - 1615 = - 1615

= \frac{- 16 15}{- 2 \cdot 32}

Multiply the numbers:

2 \cdot 32 = 64

= \frac{- 16 15}{- 64}

Apply the fraction rule:

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

= \frac{16 15}{64}

Cancel the common factor:

16

= \frac{15}{4}

The solutions to the quadratic equation are:

u = - \frac{15}{4}, u = \frac{15}{4}

u = - \frac{15}{4}, u = \frac{15}{4}

Verify Solutions

Find undefined (singularity) points:

u = - 1, u = 1

Take the denominator(s) of

\frac{1}{1 + u} + \frac{1}{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

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, u = 1

Combine undefined points with solutions:

u = - \frac{15}{4}, u = \frac{15}{4}

Substitute back

u = cos (x)

cos (x) = - \frac{15}{4}, cos (x) = \frac{15}{4}

cos (x) = - \frac{15}{4}, cos (x) = \frac{15}{4}

cos (x) = - \frac{15}{4} : x = arccos (- \frac{15}{4}) + 2 πn, x = - arccos (- \frac{15}{4}) + 2 πn

cos (x) = - \frac{15}{4}

Apply trig inverse properties

cos (x) = - \frac{15}{4}

General solutions for

cos (x) = - \frac{15}{4}

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

x = arccos (- \frac{15}{4}) + 2 πn, x = - arccos (- \frac{15}{4}) + 2 πn

x = arccos (- \frac{15}{4}) + 2 πn, x = - arccos (- \frac{15}{4}) + 2 πn

cos (x) = \frac{15}{4} : x = arccos (\frac{15}{4}) + 2 πn, x = 2 π - arccos (\frac{15}{4}) + 2 πn

cos (x) = \frac{15}{4}

Apply trig inverse properties

cos (x) = \frac{15}{4}

General solutions for

cos (x) = \frac{15}{4}

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

x = arccos (\frac{15}{4}) + 2 πn, x = 2 π - arccos (\frac{15}{4}) + 2 πn

x = arccos (\frac{15}{4}) + 2 πn, x = 2 π - arccos (\frac{15}{4}) + 2 πn

Combine all the solutions

x = arccos (- \frac{15}{4}) + 2 πn, x = - arccos (- \frac{15}{4}) + 2 πn, x = arccos (\frac{15}{4}) + 2 πn, x = 2 π - arccos (\frac{15}{4}) + 2 πn

Show solutions in decimal form

x = 2.88891 \dots + 2 πn, x = - 2.88891 \dots + 2 πn, x = 0.25268 \dots + 2 πn, x = 2 π - 0.25268 \dots + 2 πn

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