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Popular Trigonometry >

tan(x)*(2tan(x))/(1-tan^2(x))>1

Solution

tan (x) \cdot \frac{2 tan ( x )}{1 - tan ^{2} ( x )} > 1

Solution

\frac{π}{6} + πn < x < \frac{π}{4} + πn or \frac{3 π}{4} + πn < x < \frac{5 π}{6} + πn

Interval Notation

(\frac{π}{6} + πn, \frac{π}{4} + πn) \cup (\frac{3 π}{4} + πn, \frac{5 π}{6} + πn)

Decimal

0.52359 \dots + πn < x < 0.78539 \dots + πn or 2.35619 \dots + πn < x < 2.61799 \dots + πn

Solution steps

tan (x) \frac{2 tan ( x )}{1 - tan ^{2} ( x )} > 1

Let:

u = tan (x)

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

u \frac{2 u}{1 - u ^{2}} > 1 : - 1 < u < - \frac{3}{3} or \frac{3}{3} < u < 1

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

Rewrite in standard form

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

Subtract

1

from both sides

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

Simplify

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

Simplify

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

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

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

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

Multiply fractions:

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

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

2 uu = 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

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

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

\frac{2 u ^{2}}{1 - u ^{2}} - 1 > 0

Simplify

\frac{2 u ^{2}}{1 - u ^{2}} - 1 : \frac{3 u ^{2} - 1}{1 - u ^{2}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot (1 - u^{2}) = (1 - u^{2})

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

Expand

2 u^{2} - (1 - u^{2}) : 3 u^{2} - 1

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

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

- (1 - u^{2})

Distribute parentheses

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

Apply minus-plus rules

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

= - 1 + u^{2}

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

Simplify

2 u^{2} - 1 + u^{2} : 3 u^{2} - 1

2 u^{2} - 1 + u^{2}

Group like terms

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

Add similar elements:

2 u^{2} + u^{2} = 3 u^{2}

= 3 u^{2} - 1

= 3 u^{2} - 1

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

\frac{3 u ^{2} - 1}{1 - u ^{2}} > 0

\frac{3 u ^{2} - 1}{1 - u ^{2}} > 0

Factor

\frac{3 u ^{2} - 1}{1 - u ^{2}} : \frac{( 3 u + 1 ) ( 3 u - 1 )}{- ( u + 1 ) ( u - 1 )}

\frac{3 u ^{2} - 1}{1 - u ^{2}}

Factor

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

- u^{2} + 1

Factor out common term

- 1

= - (u^{2} - 1)

Factor

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

u^{2} - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (u + 1) (u - 1)

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

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

Factor

3 u^{2} - 1 : (3 u + 1) (3 u - 1)

3 u^{2} - 1

Rewrite

3 u^{2} - 1

(3 u)^{2} - 1^{2}

3 u^{2} - 1

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} u^{2} - 1

Rewrite

1

1^{2}

= (3)^{2} u^{2} - 1^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(3)^{2} u^{2} = (3 u)^{2}

= (3 u)^{2} - 1^{2}

= (3 u)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (3 u + 1) (3 u - 1)

= \frac{( 3 u + 1 ) ( 3 u - 1 )}{- ( u + 1 ) ( u - 1 )}

\frac{( 3 u + 1 ) ( 3 u - 1 )}{- ( u + 1 ) ( u - 1 )} > 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( 3 u + 1 ) ( 3 u - 1 ) ( - 1 )}{- ( u + 1 ) ( u - 1 )} < 0 \cdot (- 1)

Simplify

\frac{( 3 u + 1 ) ( 3 u - 1 )}{( u + 1 ) ( u - 1 )} < 0

Identify the intervals

Find the signs of the factors of

\frac{( 3 u + 1 ) ( 3 u - 1 )}{( u + 1 ) ( u - 1 )}

Find the signs of

3 u + 1

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

3 u + 1 = 0

Move

1

to the right side

3 u + 1 = 0

Subtract

1

from both sides

3 u + 1 - 1 = 0 - 1

Simplify

3 u = - 1

3 u = - 1

Divide both sides by

3

3 u = - 1

Divide both sides by

3

\frac{3 u}{3} = \frac{- 1}{3}

Simplify

\frac{3 u}{3} = \frac{- 1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

3 u + 1 < 0 : u < - \frac{3}{3}

3 u + 1 < 0

Move

1

to the right side

3 u + 1 < 0

Subtract

1

from both sides

3 u + 1 - 1 < 0 - 1

Simplify

3 u < - 1

3 u < - 1

Divide both sides by

3

3 u < - 1

Divide both sides by

3

\frac{3 u}{3} < \frac{- 1}{3}

Simplify

\frac{3 u}{3} < \frac{- 1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

u < - \frac{3}{3}

u < - \frac{3}{3}

u < - \frac{3}{3}

3 u + 1 > 0 : u > - \frac{3}{3}

3 u + 1 > 0

Move

1

to the right side

3 u + 1 > 0

Subtract

1

from both sides

3 u + 1 - 1 > 0 - 1

Simplify

3 u > - 1

3 u > - 1

Divide both sides by

3

3 u > - 1

Divide both sides by

3

\frac{3 u}{3} > \frac{- 1}{3}

Simplify

\frac{3 u}{3} > \frac{- 1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

u > - \frac{3}{3}

u > - \frac{3}{3}

u > - \frac{3}{3}

Find the signs of

3 u - 1

3 u - 1 = 0 : u = \frac{3}{3}

3 u - 1 = 0

Move

1

to the right side

3 u - 1 = 0

Add

1

to both sides

3 u - 1 + 1 = 0 + 1

Simplify

3 u = 1

3 u = 1

Divide both sides by

3

3 u = 1

Divide both sides by

3

\frac{3 u}{3} = \frac{1}{3}

Simplify

\frac{3 u}{3} = \frac{1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

3 u - 1 < 0 : u < \frac{3}{3}

3 u - 1 < 0

Move

1

to the right side

3 u - 1 < 0

Add

1

to both sides

3 u - 1 + 1 < 0 + 1

Simplify

3 u < 1

3 u < 1

Divide both sides by

3

3 u < 1

Divide both sides by

3

\frac{3 u}{3} < \frac{1}{3}

Simplify

\frac{3 u}{3} < \frac{1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

u < \frac{3}{3}

u < \frac{3}{3}

u < \frac{3}{3}

3 u - 1 > 0 : u > \frac{3}{3}

3 u - 1 > 0

Move

1

to the right side

3 u - 1 > 0

Add

1

to both sides

3 u - 1 + 1 > 0 + 1

Simplify

3 u > 1

3 u > 1

Divide both sides by

3

3 u > 1

Divide both sides by

3

\frac{3 u}{3} > \frac{1}{3}

Simplify

\frac{3 u}{3} > \frac{1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

u > \frac{3}{3}

u > \frac{3}{3}

u > \frac{3}{3}

Find the signs of

u + 1

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

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

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

Find the signs of

u - 1

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

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

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

Find singularity points

Find the zeros of the denominator

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

(u + 1) (u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u + 1 = 0 or u - 1 = 0

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

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

The solutions to the quadratic equation are:

u = - 1, u = 1

Summarize in a table:

3 u + 1 3 u - 1 u + 1 u - 1 \frac{( 3 u + 1 ) ( 3 u - 1 )}{( u + 1 ) ( u - 1 )} u < - 1 - - - - + u = - 1 - - 0 - Undefined - 1 < u < - \frac{3}{3} - - + - - u = - \frac{3}{3} 0 - + - 0 - \frac{3}{3} < u < \frac{3}{3} + - + - + u = \frac{3}{3} + 0 + - 0 \frac{3}{3} < u < 1 + + + - - u = 1 + + + 0 Undefined u > 1 + + + + +

Identify the intervals that satisfy the required condition:

< 0

- 1 < u < - \frac{3}{3} or \frac{3}{3} < u < 1

- 1 < u < - \frac{3}{3} or \frac{3}{3} < u < 1

- 1 < u < - \frac{3}{3} or \frac{3}{3} < u < 1

Substitute back

u = tan (x)

- 1 < tan (x) < - \frac{3}{3} or \frac{3}{3} < tan (x) < 1

- 1 < tan (x) < - \frac{3}{3} : \frac{3 π}{4} + πn < x < \frac{5 π}{6} + πn

- 1 < tan (x) < - \frac{3}{3}

a < u < b

then

a < u and u < b

- 1 < tan (x) and tan (x) < - \frac{3}{3}

- 1 < tan (x) : - \frac{π}{4} + πn < x < \frac{π}{2} + πn

- 1 < tan (x)

Switch sides

tan (x) > - 1

tan (x) > a

then

arctan (a) + πn < x < \frac{π}{2} + πn

arctan (- 1) + πn < x < \frac{π}{2} + πn

Simplify

arctan (- 1) : - \frac{π}{4}

arctan (- 1)

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1) = - arctan (1)

= - arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

arctan (1)

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{4}

= - \frac{π}{4}

- \frac{π}{4} + πn < x < \frac{π}{2} + πn

tan (x) < - \frac{3}{3} : - \frac{π}{2} + πn < x < - \frac{π}{6} + πn

tan (x) < - \frac{3}{3}

tan (x) < a

then

- \frac{π}{2} + πn < x < arctan (a) + πn

- \frac{π}{2} + πn < x < arctan (- \frac{3}{3}) + πn

Simplify

arctan (- \frac{3}{3}) : - \frac{π}{6}

arctan (- \frac{3}{3})

Use the following property:

arctan (- x) = - arctan (x)

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

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

Use the following trivial identity:

arctan (\frac{3}{3}) = \frac{π}{6}

arctan (\frac{3}{3})

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{6}

= - \frac{π}{6}

- \frac{π}{2} + πn < x < - \frac{π}{6} + πn

Combine the intervals

- \frac{π}{4} + πn < x < \frac{π}{2} + πn and - \frac{π}{2} + πn < x < - \frac{π}{6} + πn

Merge Overlapping Intervals

\frac{3 π}{4} + πn < x < \frac{5 π}{6} + πn

\frac{3}{3} < tan (x) < 1 : \frac{π}{6} + πn < x < \frac{π}{4} + πn

\frac{3}{3} < tan (x) < 1

a < u < b

then

a < u and u < b

\frac{3}{3} < tan (x) and tan (x) < 1

\frac{3}{3} < tan (x) : \frac{π}{6} + πn < x < \frac{π}{2} + πn

\frac{3}{3} < tan (x)

Switch sides

tan (x) > \frac{3}{3}

tan (x) > a

then

arctan (a) + πn < x < \frac{π}{2} + πn

arctan (\frac{3}{3}) + πn < x < \frac{π}{2} + πn

Simplify

arctan (\frac{3}{3}) : \frac{π}{6}

arctan (\frac{3}{3})

Use the following trivial identity:

arctan (\frac{3}{3}) = \frac{π}{6}

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{6}

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

tan (x) < 1 : - \frac{π}{2} + πn < x < \frac{π}{4} + πn

tan (x) < 1

tan (x) < a

then

- \frac{π}{2} + πn < x < arctan (a) + πn

- \frac{π}{2} + πn < x < arctan (1) + πn

Simplify

arctan (1) : \frac{π}{4}

arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= \frac{π}{4}

- \frac{π}{2} + πn < x < \frac{π}{4} + πn

Combine the intervals

\frac{π}{6} + πn < x < \frac{π}{2} + πn and - \frac{π}{2} + πn < x < \frac{π}{4} + πn

Merge Overlapping Intervals

\frac{π}{6} + πn < x < \frac{π}{4} + πn

Combine the intervals

\frac{3 π}{4} + πn < x < \frac{5 π}{6} + πn or \frac{π}{6} + πn < x < \frac{π}{4} + πn

Merge Overlapping Intervals

\frac{π}{6} + πn < x < \frac{π}{4} + πn or \frac{3 π}{4} + πn < x < \frac{5 π}{6} + πn

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