What is the general solution for tan(x)+cot(x)=2sqrt(2) ?

The general solution for tan(x)+cot(x)=2sqrt(2) is x=0.39269…+pin,x=1.17809…+pin

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tan(x)+cot(x)=2sqrt(2)

Solution

tan (x) + cot (x) = 22

Solution

x = 0.39269 \dots + πn, x = 1.17809 \dots + πn

Degrees

x = 22. 5^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) + cot (x) = 22

Subtract

22

from both sides

tan (x) + cot (x) - 22 = 0

Rewrite using trig identities

cot (x) + tan (x) - 22

Use the basic trigonometric identity:

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

= cot (x) + \frac{1}{cot ( x )} - 22

cot (x) + \frac{1}{cot ( x )} - 22 = 0

Solve by substitution

cot (x) + \frac{1}{cot ( x )} - 22 = 0

Let:

cot (x) = u

u + \frac{1}{u} - 22 = 0

u + \frac{1}{u} - 22 = 0 : u = 2 + 1, u = 2 - 1

u + \frac{1}{u} - 22 = 0

Multiply both sides by

u

u + \frac{1}{u} - 22 = 0

Multiply both sides by

u

uu + \frac{1}{u} u - 22 u = 0 \cdot u

Simplify

uu + \frac{1}{u} u - 22 u = 0 \cdot u

Simplify

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}

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

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

Cancel the common factor:

u

= 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} + 1 - 22 u = 0

u^{2} + 1 - 22 u = 0

u^{2} + 1 - 22 u = 0

Solve

u^{2} + 1 - 22 u = 0 : u = 2 + 1, u = 2 - 1

u^{2} + 1 - 22 u = 0

Write in the standard form

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

u^{2} - 22 u + 1 = 0

Solve with the quadratic formula

u^{2} - 22 u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 22, c = 1

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

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

(- 22)^{2} - 4 \cdot 1 \cdot 1 = 2

(- 22)^{2} - 4 \cdot 1 \cdot 1

(- 22)^{2} = 2^{3}

(- 22)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 22)^{2} = (22)^{2}

= (22)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (2)^{2}

(2)^{2} : 2

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

= 2^{2} \cdot 2

Apply exponent rule:

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

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

= 2^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2^{3}

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 4

= 2^{3} - 4

2^{3} = 8

= 8 - 4

Subtract the numbers:

8 - 4 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

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

Separate the solutions

u_{1} = \frac{- ( - 2 2 ) + 2}{2 \cdot 1}, u_{2} = \frac{- ( - 2 2 ) - 2}{2 \cdot 1}

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 2 + 2}{2}

Factor

22 + 2 : 2 (2 + 1)

22 + 2

Rewrite as

= 22 + 2 \cdot 1

Factor out common term

2

= 2 (2 + 1)

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

Divide the numbers:

\frac{2}{2} = 1

= 2 + 1

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 2 - 2}{2}

Factor

22 - 2 : 2 (2 - 1)

22 - 2

Rewrite as

= 22 - 2 \cdot 1

Factor out common term

2

= 2 (2 - 1)

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

Divide the numbers:

\frac{2}{2} = 1

= 2 - 1

The solutions to the quadratic equation are:

u = 2 + 1, u = 2 - 1

u = 2 + 1, u = 2 - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u + \frac{1}{u} - 22

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 2 + 1, u = 2 - 1

Substitute back

u = cot (x)

cot (x) = 2 + 1, cot (x) = 2 - 1

cot (x) = 2 + 1, cot (x) = 2 - 1

cot (x) = 2 + 1 : x = arccot (2 + 1) + πn

cot (x) = 2 + 1

Apply trig inverse properties

cot (x) = 2 + 1

General solutions for

cot (x) = 2 + 1

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (2 + 1) + πn

x = arccot (2 + 1) + πn

cot (x) = 2 - 1 : x = arccot (2 - 1) + πn

cot (x) = 2 - 1

Apply trig inverse properties

cot (x) = 2 - 1

General solutions for

cot (x) = 2 - 1

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (2 - 1) + πn

x = arccot (2 - 1) + πn

Combine all the solutions

x = arccot (2 + 1) + πn, x = arccot (2 - 1) + πn

Show solutions in decimal form

x = 0.39269 \dots + πn, x = 1.17809 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)+cot(x)=2sqrt(2) ?
The general solution for tan(x)+cot(x)=2sqrt(2) is x=0.39269…+pin,x=1.17809…+pin