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Popular >

(cos^2(x))/(1-cos(x))<0

Solution

\frac{cos ^{2} ( x )}{1 - cos ( x )} < 0

Solution

False for all x \in R

Solution steps

\frac{cos ^{2} ( x )}{1 - cos ( x )} < 0

Let:

u = cos (x)

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

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

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

Identify the intervals

Find the signs of the factors of

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

Find the signs of

u^{2}

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

u^{2} > 0 : u < 0 or u > 0

u^{2} > 0

For

u^{n} > 0

, if

n

is even then

u < 0

u > 0

u < 0 or u > 0

Find the signs of

1 - u

1 - u = 0 : u = 1

1 - u = 0

Move

1

to the right side

1 - u = 0

Subtract

1

from both sides

1 - u - 1 = 0 - 1

Simplify

- u = - 1

- u = - 1

Divide both sides by

- 1

- u = - 1

Divide both sides by

- 1

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

Simplify

u = 1

u = 1

1 - u < 0 : u > 1

1 - u < 0

Move

1

to the right side

1 - u < 0

Subtract

1

from both sides

1 - u - 1 < 0 - 1

Simplify

- u < - 1

- u < - 1

Multiply both sides by

- 1

- u < - 1

Multiply both sides by -1 (reverse the inequality)

(- u) (- 1) > (- 1) (- 1)

Simplify

u > 1

u > 1

1 - u > 0 : u < 1

1 - u > 0

Move

1

to the right side

1 - u > 0

Subtract

1

from both sides

1 - u - 1 > 0 - 1

Simplify

- u > - 1

- u > - 1

Multiply both sides by

- 1

- u > - 1

Multiply both sides by -1 (reverse the inequality)

(- u) (- 1) < (- 1) (- 1)

Simplify

u < 1

u < 1

Find singularity points

Find the zeros of the denominator

1 - u : u = 1

1 - u = 0

Move

1

to the right side

1 - u = 0

Subtract

1

from both sides

1 - u - 1 = 0 - 1

Simplify

- u = - 1

- u = - 1

Divide both sides by

- 1

- u = - 1

Divide both sides by

- 1

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

Simplify

u = 1

u = 1

Summarize in a table:

u^{2} 1 - u \frac{u ^{2}}{1 - u} u < 0 + + + u = 0 0 + 0 0 < u < 1 + + + u = 1 + 0 Undefined u > 1 + - -

Identify the intervals that satisfy the required condition:

< 0

u > 1

u > 1

u > 1

Substitute back

u = cos (x)

cos (x) > 1

Range of

cos (x) : - 1 \leq cos (x) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x) \leq 1

- 1 \leq cos (x) \leq 1

cos (x) > 1 and - 1 \leq cos (x) \leq 1 :

False

Let

y = cos (x)

Combine the intervals

y > 1 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y > 1 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y > 1

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

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