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1/(sec(2x)-1)-1/(sec(2x)+1)=6

Solution

\frac{1}{sec ( 2 x ) - 1} - \frac{1}{sec ( 2 x ) + 1} = 6

Solution

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn, x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

Degrees

x = 1 5^{\circ} + 18 0^{\circ} n, x = 16 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n, x = 10 5^{\circ} + 18 0^{\circ} n

Solution steps

\frac{1}{sec ( 2 x ) - 1} - \frac{1}{sec ( 2 x ) + 1} = 6

Solve by substitution

\frac{1}{sec ( 2 x ) - 1} - \frac{1}{sec ( 2 x ) + 1} = 6

Let:

sec (2 x) = u

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

\frac{1}{u - 1} - \frac{1}{u + 1} = 6 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

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

Multiply by LCM

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

Find Least Common Multiplier of

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

u - 1, u + 1

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u - 1

u + 1

= (u - 1) (u + 1)

Multiply by LCM=

(u - 1) (u + 1)

\frac{1}{u - 1} (u - 1) (u + 1) - \frac{1}{u + 1} (u - 1) (u + 1) = 6 (u - 1) (u + 1)

Simplify

\frac{1}{u - 1} (u - 1) (u + 1) - \frac{1}{u + 1} (u - 1) (u + 1) = 6 (u - 1) (u + 1)

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

u - 1

= 1 \cdot (u + 1)

Refine

= u + 1

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

u + 1

= - 1 \cdot (u - 1)

Multiply:

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

= - (u - 1)

u + 1 - (u - 1) = 6 (u - 1) (u + 1)

u + 1 - (u - 1) = 6 (u - 1) (u + 1)

u + 1 - (u - 1) = 6 (u - 1) (u + 1)

Solve

u + 1 - (u - 1) = 6 (u - 1) (u + 1) : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

u + 1 - (u - 1) = 6 (u - 1) (u + 1)

Expand

u + 1 - (u - 1) : 2

u + 1 - (u - 1)

- (u - 1) : - u + 1

- (u - 1)

Distribute parentheses

= - (u) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - u + 1

= u + 1 - u + 1

Simplify

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

Expand

6 (u - 1) (u + 1) : 6 u^{2} - 6

6 (u - 1) (u + 1)

Expand

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

(u - 1) (u + 1)

Apply Difference of Two Squares Formula:

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

a = u, b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= u^{2} - 1

= 6 (u^{2} - 1)

Expand

6 (u^{2} - 1) : 6 u^{2} - 6

6 (u^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 6, b = u^{2}, c = 1

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

Multiply the numbers:

6 \cdot 1 = 6

= 6 u^{2} - 6

= 6 u^{2} - 6

2 = 6 u^{2} - 6

Switch sides

6 u^{2} - 6 = 2

Move

6

to the right side

6 u^{2} - 6 = 2

Add

6

to both sides

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

Simplify

6 u^{2} = 8

6 u^{2} = 8

Divide both sides by

6

6 u^{2} = 8

Divide both sides by

6

\frac{6 u ^{2}}{6} = \frac{8}{6}

Simplify

u^{2} = \frac{4}{3}

u^{2} = \frac{4}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{4}{3}, u = - \frac{4}{3}

\frac{4}{3} = \frac{2 3}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

Rationalize

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

- \frac{4}{3} = - \frac{2 3}{3}

- \frac{4}{3}

Simplify

\frac{4}{3} : \frac{2}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

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

and compare to zero

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

Substitute back

u = sec (2 x)

sec (2 x) = \frac{2 3}{3}, sec (2 x) = - \frac{2 3}{3}

sec (2 x) = \frac{2 3}{3}, sec (2 x) = - \frac{2 3}{3}

sec (2 x) = \frac{2 3}{3} : x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn

sec (2 x) = \frac{2 3}{3}

General solutions for

sec (2 x) = \frac{2 3}{3}

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{11 π}{6} + 2 πn

Solve

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

Solve

2 x = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{12} + πn

2 x = \frac{11 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{11 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2} : \frac{11 π}{12} + πn

\frac{\frac{11 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{11 π}{6}}{2} = \frac{11 π}{12}

\frac{\frac{11 π}{6}}{2}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{11 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{11 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn

sec (2 x) = - \frac{2 3}{3} : x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

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

General solutions for

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

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

2 x = \frac{5 π}{6} + 2 πn, 2 x = \frac{7 π}{6} + 2 πn

2 x = \frac{5 π}{6} + 2 πn, 2 x = \frac{7 π}{6} + 2 πn

Solve

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{5 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

Solve

2 x = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{12} + πn

2 x = \frac{7 π}{6} + 2 πn

Divide both sides by

2

2 x = \frac{7 π}{6} + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} = \frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2} : \frac{7 π}{12} + πn

\frac{\frac{7 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{6}}{2} = \frac{7 π}{12}

\frac{\frac{7 π}{6}}{2}

Apply the fraction rule:

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

= \frac{7 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{7 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{7 π}{12} + πn

x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

Combine all the solutions

x = \frac{π}{12} + πn, x = \frac{11 π}{12} + πn, x = \frac{5 π}{12} + πn, x = \frac{7 π}{12} + πn

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