What is the general solution for tan(x)+cot(x)=6,sin^6(x)+cos^6(x) ?

The general solution for tan(x)+cot(x)=6,sin^6(x)+cos^6(x) is No Solution for x\in\mathbb{R}

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tan(x)+cot(x)=6,sin^6(x)+cos^6(x)

Solution

tan (x) + cot (x) = 6, sin^{6} (x) + cos^{6} (x)

Solution

No Solution for x \in R

Solution steps

tan (x) + cot (x) = 6, sin^{6} (x) + cos^{6} (x)

Subtract

6

from both sides

tan (x) + cot (x) - 6 = 0

Rewrite using trig identities

- 6 + cot (x) + tan (x)

Use the basic trigonometric identity:

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

= - 6 + cot (x) + \frac{1}{cot ( x )}

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

Solve by substitution

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

Let:

cot (x) = u

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

- 6 + u + \frac{1}{u} = 0 : u = 3 + 22, u = 3 - 22

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

- 6 u + uu + \frac{1}{u} 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

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

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

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

Solve

- 6 u + u^{2} + 1 = 0 : u = 3 + 22, u = 3 - 22

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

Write in the standard form

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

u^{2} - 6 u + 1 = 0

Solve with the quadratic formula

u^{2} - 6 u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 6, c = 1

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

(- 6)^{2} - 4 \cdot 1 \cdot 1 = 42

(- 6)^{2} - 4 \cdot 1 \cdot 1

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 6^{2} - 4

6^{2} = 36

= 36 - 4

Subtract the numbers:

36 - 4 = 32

= 32

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

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

2

is a prime number, therefore no further factorization is possible

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

= 2^{5}

= 2^{5}

Apply exponent rule:

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

= 2^{4} \cdot 2

Apply radical rule:

= 2 2^{4}

Apply radical rule:

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

= 2^{2} 2

Refine

= 42

u_{1, 2} = \frac{- ( - 6 ) \pm 4 2}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 4 2}{2 \cdot 1}, u_{2} = \frac{- ( - 6 ) - 4 2}{2 \cdot 1}

u = \frac{- ( - 6 ) + 4 2}{2 \cdot 1} : 3 + 22

\frac{- ( - 6 ) + 4 2}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 + 4 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{6 + 4 2}{2}

Factor

6 + 42 : 2 (3 + 22)

6 + 42

Rewrite as

= 2 \cdot 3 + 2 \cdot 22

Factor out common term

2

= 2 (3 + 22)

= \frac{2 ( 3 + 2 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 + 22

u = \frac{- ( - 6 ) - 4 2}{2 \cdot 1} : 3 - 22

\frac{- ( - 6 ) - 4 2}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 - 4 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{6 - 4 2}{2}

Factor

6 - 42 : 2 (3 - 22)

6 - 42

Rewrite as

= 2 \cdot 3 - 2 \cdot 22

Factor out common term

2

= 2 (3 - 22)

= \frac{2 ( 3 - 2 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 - 22

The solutions to the quadratic equation are:

u = 3 + 22, u = 3 - 22

u = 3 + 22, u = 3 - 22

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3 + 22, u = 3 - 22

Substitute back

u = cot (x)

cot (x) = 3 + 22, cot (x) = 3 - 22

cot (x) = 3 + 22, cot (x) = 3 - 22

cot (x) = 3 + 22, sin^{6} (x) + cos^{6} (x) :

No Solution

cot (x) = 3 + 22, sin^{6} (x) + cos^{6} (x)

Apply trig inverse properties

cot (x) = 3 + 22

General solutions for

cot (x) = 3 + 22

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

x = arccot (3 + 22) + πn

x = arccot (3 + 22) + πn

Solutions for the range

sin^{6} (x) + cos^{6} (x)

No Solution

cot (x) = 3 - 22, sin^{6} (x) + cos^{6} (x) :

No Solution

cot (x) = 3 - 22, sin^{6} (x) + cos^{6} (x)

Apply trig inverse properties

cot (x) = 3 - 22

General solutions for

cot (x) = 3 - 22

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

x = arccot (3 - 22) + πn

x = arccot (3 - 22) + πn

Solutions for the range

sin^{6} (x) + cos^{6} (x)

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)+cot(x)=6,sin^6(x)+cos^6(x) ?
The general solution for tan(x)+cot(x)=6,sin^6(x)+cos^6(x) is No Solution for x\in\mathbb{R}