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(tan(x)+1)^2+(1+1/(tan(x)))^2=100

Solution

(tan (x) + 1)^{2} + (1 + \frac{1}{tan ( x )})^{2} = 100

Solution

x = - 0.09100 \dots + πn, x = 0.11141 \dots + πn, x = 1.45937 \dots + πn, x = - 1.47979 \dots + πn

Degrees

x = - 5.21393 \dots^{\circ} + 18 0^{\circ} n, x = 6.38381 \dots^{\circ} + 18 0^{\circ} n, x = 83.61618 \dots^{\circ} + 18 0^{\circ} n, x = - 84.78606 \dots^{\circ} + 18 0^{\circ} n

Solution steps

(tan (x) + 1)^{2} + (1 + \frac{1}{tan ( x )})^{2} = 100

Solve by substitution

(tan (x) + 1)^{2} + (1 + \frac{1}{tan ( x )})^{2} = 100

Let:

tan (x) = u

(u + 1)^{2} + (1 + \frac{1}{u})^{2} = 100

(u + 1)^{2} + (1 + \frac{1}{u})^{2} = 100 : u \approx - 0.09125 \dots, u \approx 0.11188 \dots, u \approx 8.93799 \dots, u \approx - 10.95862 \dots

(u + 1)^{2} + (1 + \frac{1}{u})^{2} = 100

Multiply by LCM

(u + 1)^{2} + (1 + \frac{1}{u})^{2} = 100

Expand

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

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

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

Apply Perfect Square Formula:

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

a = u, b = 1

= u^{2} + 2 u \cdot 1 + 1^{2}

Simplify

u^{2} + 2 u \cdot 1 + 1^{2} : u^{2} + 2 u + 1

u^{2} + 2 u \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= u^{2} + 2 \cdot 1 \cdot u + 1

Multiply the numbers:

2 \cdot 1 = 2

= u^{2} + 2 u + 1

= u^{2} + 2 u + 1

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

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

Apply Perfect Square Formula:

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

a = 1, b = \frac{1}{u}

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

2 \cdot 1 \cdot \frac{1}{u}

Multiply fractions:

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

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

Refine

= \frac{2}{u}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

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

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

Add the numbers:

1 + 1 = 2

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

u^{2} + 2 u + \frac{2}{u} + \frac{1}{u ^{2}} + 2 = 100

Move

2

to the right side

u^{2} + 2 u + \frac{2}{u} + \frac{1}{u ^{2}} + 2 = 100

Subtract

2

from both sides

u^{2} + 2 u + \frac{2}{u} + \frac{1}{u ^{2}} + 2 - 2 = 100 - 2

Simplify

u^{2} + 2 u + \frac{2}{u} + \frac{1}{u ^{2}} = 98

u^{2} + 2 u + \frac{2}{u} + \frac{1}{u ^{2}} = 98

Find Least Common Multiplier of

u, u^{2} : u^{2}

u, u^{2}

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u

u^{2}

= u^{2}

Multiply by LCM=

u^{2}

u^{2} u^{2} + 2 u u^{2} + \frac{2}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 98 u^{2}

Simplify

u^{2} u^{2} + 2 u u^{2} + \frac{2}{u} u^{2} + \frac{1}{u ^{2}} u^{2} = 98 u^{2}

Simplify

u^{2} u^{2} : u^{4}

u^{2} u^{2}

Apply exponent rule:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

Simplify

2 u u^{2} : 2 u^{3}

2 u u^{2}

Apply exponent rule:

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

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

= 2 u^{1 + 2}

Add the numbers:

1 + 2 = 3

= 2 u^{3}

Simplify

\frac{2}{u} u^{2} : 2 u

\frac{2}{u} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u

= 2 u

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

u^{2}

= 1

u^{4} + 2 u^{3} + 2 u + 1 = 98 u^{2}

u^{4} + 2 u^{3} + 2 u + 1 = 98 u^{2}

u^{4} + 2 u^{3} + 2 u + 1 = 98 u^{2}

Solve

u^{4} + 2 u^{3} + 2 u + 1 = 98 u^{2} : u \approx - 0.09125 \dots, u \approx 0.11188 \dots, u \approx 8.93799 \dots, u \approx - 10.95862 \dots

u^{4} + 2 u^{3} + 2 u + 1 = 98 u^{2}

Move

98 u^{2}

to the left side

u^{4} + 2 u^{3} + 2 u + 1 = 98 u^{2}

Subtract

98 u^{2}

from both sides

u^{4} + 2 u^{3} + 2 u + 1 - 98 u^{2} = 98 u^{2} - 98 u^{2}

Simplify

u^{4} + 2 u^{3} + 2 u + 1 - 98 u^{2} = 0

u^{4} + 2 u^{3} + 2 u + 1 - 98 u^{2} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} + 2 u^{3} - 98 u^{2} + 2 u + 1 = 0

Find one solution for

u^{4} + 2 u^{3} - 98 u^{2} + 2 u + 1 = 0

using Newton-Raphson:

u \approx - 0.09125 \dots

u^{4} + 2 u^{3} - 98 u^{2} + 2 u + 1 = 0

Newton-Raphson Approximation Definition

f (u) = u^{4} + 2 u^{3} - 98 u^{2} + 2 u + 1

Find

f^{^{'}} (u) : 4 u^{3} + 6 u^{2} - 196 u + 2

\frac{d}{d u} (u^{4} + 2 u^{3} - 98 u^{2} + 2 u + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{4}) + \frac{d}{d u} (2 u^{3}) - \frac{d}{d u} (98 u^{2}) + \frac{d}{d u} (2 u) + \frac{d}{d u} (1)

\frac{d}{d u} (u^{4}) = 4 u^{3}

\frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 u^{4 - 1}

Simplify

= 4 u^{3}

\frac{d}{d u} (2 u^{3}) = 6 u^{2}

\frac{d}{d u} (2 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 \cdot 3 u^{3 - 1}

Simplify

= 6 u^{2}

\frac{d}{d u} (98 u^{2}) = 196 u

\frac{d}{d u} (98 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 98 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 98 \cdot 2 u^{2 - 1}

Simplify

= 196 u

\frac{d}{d u} (2 u) = 2

\frac{d}{d u} (2 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 2 \cdot 1

Simplify

= 2

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 4 u^{3} + 6 u^{2} - 196 u + 2 + 0

Simplify

= 4 u^{3} + 6 u^{2} - 196 u + 2

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.5 : Δ u_{1} = 0.5

f (u_{0}) = 0^{4} + 2 \cdot 0^{3} - 98 \cdot 0^{2} + 2 \cdot 0 + 1 = 1

f^{^{'}} (u_{0}) = 4 \cdot 0^{3} + 6 \cdot 0^{2} - 196 \cdot 0 + 2 = 2

u_{1} = - 0.5

Δ u_{1} = ∣ - 0.5 - 0 ∣ = 0.5

Δ u_{1} = 0.5

u_{2} = - 0.25556 \dots : Δ u_{2} = 0.24443 \dots

f (u_{1}) = (- 0.5)^{4} + 2 (- 0.5)^{3} - 98 (- 0.5)^{2} + 2 (- 0.5) + 1 = - 24.6875

f^{^{'}} (u_{1}) = 4 (- 0.5)^{3} + 6 (- 0.5)^{2} - 196 (- 0.5) + 2 = 101

u_{2} = - 0.25556 \dots

Δ u_{2} = ∣ - 0.25556 \dots - (- 0.5) ∣ = 0.24443 \dots

Δ u_{2} = 0.24443 \dots

u_{3} = - 0.14222 \dots : Δ u_{3} = 0.11334 \dots

f (u_{2}) = (- 0.25556 \dots)^{4} + 2 (- 0.25556 \dots)^{3} - 98 (- 0.25556 \dots)^{2} + 2 (- 0.25556 \dots) + 1 = - 5.94119 \dots

f^{^{'}} (u_{2}) = 4 (- 0.25556 \dots)^{3} + 6 (- 0.25556 \dots)^{2} - 196 (- 0.25556 \dots) + 2 = 52.41670 \dots

u_{3} = - 0.14222 \dots

Δ u_{3} = ∣ - 0.14222 \dots - (- 0.25556 \dots) ∣ = 0.11334 \dots

Δ u_{3} = 0.11334 \dots

u_{4} = - 0.09980 \dots : Δ u_{4} = 0.04242 \dots

f (u_{3}) = (- 0.14222 \dots)^{4} + 2 (- 0.14222 \dots)^{3} - 98 (- 0.14222 \dots)^{2} + 2 (- 0.14222 \dots) + 1 = - 1.27210 \dots

f^{^{'}} (u_{3}) = 4 (- 0.14222 \dots)^{3} + 6 (- 0.14222 \dots)^{2} - 196 (- 0.14222 \dots) + 2 = 29.98574 \dots

u_{4} = - 0.09980 \dots

Δ u_{4} = ∣ - 0.09980 \dots - (- 0.14222 \dots) ∣ = 0.04242 \dots

Δ u_{4} = 0.04242 \dots

u_{5} = - 0.09158 \dots : Δ u_{5} = 0.00821 \dots

f (u_{4}) = (- 0.09980 \dots)^{4} + 2 (- 0.09980 \dots)^{3} - 98 (- 0.09980 \dots)^{2} + 2 (- 0.09980 \dots) + 1 = - 0.17758 \dots

f^{^{'}} (u_{4}) = 4 (- 0.09980 \dots)^{3} + 6 (- 0.09980 \dots)^{2} - 196 (- 0.09980 \dots) + 2 = 21.61665 \dots

u_{5} = - 0.09158 \dots

Δ u_{5} = ∣ - 0.09158 \dots - (- 0.09980 \dots) ∣ = 0.00821 \dots

Δ u_{5} = 0.00821 \dots

u_{6} = - 0.09125 \dots : Δ u_{6} = 0.00033 \dots

f (u_{5}) = (- 0.09158 \dots)^{4} + 2 (- 0.09158 \dots)^{3} - 98 (- 0.09158 \dots)^{2} + 2 (- 0.09158 \dots) + 1 = - 0.00664 \dots

f^{^{'}} (u_{5}) = 4 (- 0.09158 \dots)^{3} + 6 (- 0.09158 \dots)^{2} - 196 (- 0.09158 \dots) + 2 = 19.99798 \dots

u_{6} = - 0.09125 \dots

Δ u_{6} = ∣ - 0.09125 \dots - (- 0.09158 \dots) ∣ = 0.00033 \dots

Δ u_{6} = 0.00033 \dots

u_{7} = - 0.09125 \dots : Δ u_{7} = 5.46294 E - 7

f (u_{6}) = (- 0.09125 \dots)^{4} + 2 (- 0.09125 \dots)^{3} - 98 (- 0.09125 \dots)^{2} + 2 (- 0.09125 \dots) + 1 = - 0.00001 \dots

f^{^{'}} (u_{6}) = 4 (- 0.09125 \dots)^{3} + 6 (- 0.09125 \dots)^{2} - 196 (- 0.09125 \dots) + 2 = 19.93248 \dots

u_{7} = - 0.09125 \dots

Δ u_{7} = ∣ - 0.09125 \dots - (- 0.09125 \dots) ∣ = 5.46294 E - 7

Δ u_{7} = 5.46294 E - 7

u \approx - 0.09125 \dots

Apply long division:

\frac{u ^{4} + 2 u ^{3} - 98 u ^{2} + 2 u + 1}{u + 0.09125 \dots} = u^{3} + 1.90874 \dots u^{2} - 98.17417 \dots u + 10.95862 \dots

u^{3} + 1.90874 \dots u^{2} - 98.17417 \dots u + 10.95862 \dots \approx 0

Find one solution for

u^{3} + 1.90874 \dots u^{2} - 98.17417 \dots u + 10.95862 \dots = 0

using Newton-Raphson:

u \approx 0.11188 \dots

u^{3} + 1.90874 \dots u^{2} - 98.17417 \dots u + 10.95862 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} + 1.90874 \dots u^{2} - 98.17417 \dots u + 10.95862 \dots

Find

f^{^{'}} (u) : 3 u^{2} + 3.81749 \dots u - 98.17417 \dots

\frac{d}{d u} (u^{3} + 1.90874 \dots u^{2} - 98.17417 \dots u + 10.95862 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{3}) + \frac{d}{d u} (1.90874 \dots u^{2}) - \frac{d}{d u} (98.17417 \dots u) + \frac{d}{d u} (10.95862 \dots)

\frac{d}{d u} (u^{3}) = 3 u^{2}

\frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

\frac{d}{d u} (1.90874 \dots u^{2}) = 3.81749 \dots u

\frac{d}{d u} (1.90874 \dots u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 1.90874 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 1.90874 \dots \cdot 2 u^{2 - 1}

Simplify

= 3.81749 \dots u

\frac{d}{d u} (98.17417 \dots u) = 98.17417 \dots

\frac{d}{d u} (98.17417 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 98.17417 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 98.17417 \dots \cdot 1

Simplify

= 98.17417 \dots

\frac{d}{d u} (10.95862 \dots) = 0

\frac{d}{d u} (10.95862 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 3 u^{2} + 3.81749 \dots u - 98.17417 \dots + 0

Simplify

= 3 u^{2} + 3.81749 \dots u - 98.17417 \dots

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.11162 \dots : Δ u_{1} = 0.11162 \dots

f (u_{0}) = 0^{3} + 1.90874 \dots \cdot 0^{2} - 98.17417 \dots \cdot 0 + 10.95862 \dots = 10.95862 \dots

f^{^{'}} (u_{0}) = 3 \cdot 0^{2} + 3.81749 \dots \cdot 0 - 98.17417 \dots = - 98.17417 \dots

u_{1} = 0.11162 \dots

Δ u_{1} = ∣ 0.11162 \dots - 0 ∣ = 0.11162 \dots

Δ u_{1} = 0.11162 \dots

u_{2} = 0.11188 \dots : Δ u_{2} = 0.00025 \dots

f (u_{1}) = 0.11162 \dots^{3} + 1.90874 \dots \cdot 0.11162 \dots^{2} - 98.17417 \dots \cdot 0.11162 \dots + 10.95862 \dots = 0.02517 \dots

f^{^{'}} (u_{1}) = 3 \cdot 0.11162 \dots^{2} + 3.81749 \dots \cdot 0.11162 \dots - 98.17417 \dots = - 97.71067 \dots

u_{2} = 0.11188 \dots

Δ u_{2} = ∣ 0.11188 \dots - 0.11162 \dots ∣ = 0.00025 \dots

Δ u_{2} = 0.00025 \dots

u_{3} = 0.11188 \dots : Δ u_{3} = 1.52432 E - 9

f (u_{2}) = 0.11188 \dots^{3} + 1.90874 \dots \cdot 0.11188 \dots^{2} - 98.17417 \dots \cdot 0.11188 \dots + 10.95862 \dots = 1.48941 E - 7

f^{^{'}} (u_{2}) = 3 \cdot 0.11188 \dots^{2} + 3.81749 \dots \cdot 0.11188 \dots - 98.17417 \dots = - 97.70951 \dots

u_{3} = 0.11188 \dots

Δ u_{3} = ∣ 0.11188 \dots - 0.11188 \dots ∣ = 1.52432 E - 9

Δ u_{3} = 1.52432 E - 9

u \approx 0.11188 \dots

Apply long division:

\frac{u ^{3} + 1.90874 \dots u ^{2} - 98.17417 \dots u + 10.95862 \dots}{u - 0.11188 \dots} = u^{2} + 2.02062 \dots u - 97.94810 \dots

u^{2} + 2.02062 \dots u - 97.94810 \dots \approx 0

Find one solution for

u^{2} + 2.02062 \dots u - 97.94810 \dots = 0

using Newton-Raphson:

u \approx 8.93799 \dots

u^{2} + 2.02062 \dots u - 97.94810 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} + 2.02062 \dots u - 97.94810 \dots

Find

f^{^{'}} (u) : 2 u + 2.02062 \dots

\frac{d}{d u} (u^{2} + 2.02062 \dots u - 97.94810 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{2}) + \frac{d}{d u} (2.02062 \dots u) - \frac{d}{d u} (97.94810 \dots)

\frac{d}{d u} (u^{2}) = 2 u

\frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d}{d u} (2.02062 \dots u) = 2.02062 \dots

\frac{d}{d u} (2.02062 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2.02062 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 2.02062 \dots \cdot 1

Simplify

= 2.02062 \dots

\frac{d}{d u} (97.94810 \dots) = 0

\frac{d}{d u} (97.94810 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 2 u + 2.02062 \dots - 0

Simplify

= 2 u + 2.02062 \dots

Let

u_{0} = 5

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 10.22809 \dots : Δ u_{1} = 5.22809 \dots

f (u_{0}) = 5^{2} + 2.02062 \dots \cdot 5 - 97.94810 \dots = - 62.84495 \dots

f^{^{'}} (u_{0}) = 2 \cdot 5 + 2.02062 \dots = 12.02062 \dots

u_{1} = 10.22809 \dots

Δ u_{1} = ∣ 10.22809 \dots - 5 ∣ = 5.22809 \dots

Δ u_{1} = 5.22809 \dots

u_{2} = 9.01204 \dots : Δ u_{2} = 1.21605 \dots

f (u_{1}) = 10.22809 \dots^{2} + 2.02062 \dots \cdot 10.22809 \dots - 97.94810 \dots = 27.33294 \dots

f^{^{'}} (u_{1}) = 2 \cdot 10.22809 \dots + 2.02062 \dots = 22.47681 \dots

u_{2} = 9.01204 \dots

Δ u_{2} = ∣ 9.01204 \dots - 10.22809 \dots ∣ = 1.21605 \dots

Δ u_{2} = 1.21605 \dots

u_{3} = 8.93826 \dots : Δ u_{3} = 0.07377 \dots

f (u_{2}) = 9.01204 \dots^{2} + 2.02062 \dots \cdot 9.01204 \dots - 97.94810 \dots = 1.47877 \dots

f^{^{'}} (u_{2}) = 2 \cdot 9.01204 \dots + 2.02062 \dots = 20.04471 \dots

u_{3} = 8.93826 \dots

Δ u_{3} = ∣ 8.93826 \dots - 9.01204 \dots ∣ = 0.07377 \dots

Δ u_{3} = 0.07377 \dots

u_{4} = 8.93799 \dots : Δ u_{4} = 0.00027 \dots

f (u_{3}) = 8.93826 \dots^{2} + 2.02062 \dots \cdot 8.93826 \dots - 97.94810 \dots = 0.00544 \dots

f^{^{'}} (u_{3}) = 2 \cdot 8.93826 \dots + 2.02062 \dots = 19.89716 \dots

u_{4} = 8.93799 \dots

Δ u_{4} = ∣ 8.93799 \dots - 8.93826 \dots ∣ = 0.00027 \dots

Δ u_{4} = 0.00027 \dots

u_{5} = 8.93799 \dots : Δ u_{5} = 3.76056 E - 9

f (u_{4}) = 8.93799 \dots^{2} + 2.02062 \dots \cdot 8.93799 \dots - 97.94810 \dots = 7.48225 E - 8

f^{^{'}} (u_{4}) = 2 \cdot 8.93799 \dots + 2.02062 \dots = 19.89661 \dots

u_{5} = 8.93799 \dots

Δ u_{5} = ∣ 8.93799 \dots - 8.93799 \dots ∣ = 3.76056 E - 9

Δ u_{5} = 3.76056 E - 9

u \approx 8.93799 \dots

Apply long division:

\frac{u ^{2} + 2.02062 \dots u - 97.94810 \dots}{u - 8.93799 \dots} = u + 10.95862 \dots

u + 10.95862 \dots \approx 0

u \approx - 10.95862 \dots

The solutions are

u \approx - 0.09125 \dots, u \approx 0.11188 \dots, u \approx 8.93799 \dots, u \approx - 10.95862 \dots

u \approx - 0.09125 \dots, u \approx 0.11188 \dots, u \approx 8.93799 \dots, u \approx - 10.95862 \dots

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx - 0.09125 \dots, u \approx 0.11188 \dots, u \approx 8.93799 \dots, u \approx - 10.95862 \dots

Substitute back

u = tan (x)

tan (x) \approx - 0.09125 \dots, tan (x) \approx 0.11188 \dots, tan (x) \approx 8.93799 \dots, tan (x) \approx - 10.95862 \dots

tan (x) \approx - 0.09125 \dots, tan (x) \approx 0.11188 \dots, tan (x) \approx 8.93799 \dots, tan (x) \approx - 10.95862 \dots

tan (x) = - 0.09125 \dots : x = arctan (- 0.09125 \dots) + πn

tan (x) = - 0.09125 \dots

Apply trig inverse properties

tan (x) = - 0.09125 \dots

General solutions for

tan (x) = - 0.09125 \dots

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 0.09125 \dots) + πn

x = arctan (- 0.09125 \dots) + πn

tan (x) = 0.11188 \dots : x = arctan (0.11188 \dots) + πn

tan (x) = 0.11188 \dots

Apply trig inverse properties

tan (x) = 0.11188 \dots

General solutions for

tan (x) = 0.11188 \dots

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (0.11188 \dots) + πn

x = arctan (0.11188 \dots) + πn

tan (x) = 8.93799 \dots : x = arctan (8.93799 \dots) + πn

tan (x) = 8.93799 \dots

Apply trig inverse properties

tan (x) = 8.93799 \dots

General solutions for

tan (x) = 8.93799 \dots

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (8.93799 \dots) + πn

x = arctan (8.93799 \dots) + πn

tan (x) = - 10.95862 \dots : x = arctan (- 10.95862 \dots) + πn

tan (x) = - 10.95862 \dots

Apply trig inverse properties

tan (x) = - 10.95862 \dots

General solutions for

tan (x) = - 10.95862 \dots

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 10.95862 \dots) + πn

x = arctan (- 10.95862 \dots) + πn

Combine all the solutions

x = arctan (- 0.09125 \dots) + πn, x = arctan (0.11188 \dots) + πn, x = arctan (8.93799 \dots) + πn, x = arctan (- 10.95862 \dots) + πn

Show solutions in decimal form

x = - 0.09100 \dots + πn, x = 0.11141 \dots + πn, x = 1.45937 \dots + πn, x = - 1.47979 \dots + πn

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