What is the general solution for (cot(x))/(cos(x))+1/(sin(x))=1 ?

The general solution for (cot(x))/(cos(x))+1/(sin(x))=1 is No Solution for x\in\mathbb{R}

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(cot(x))/(cos(x))+1/(sin(x))=1

Solution

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} = 1

Solution

No Solution for x \in R

Solution steps

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} = 1

Subtract

1

from both sides

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

Simplify

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} - 1 : \frac{cot ( x ) sin ( x ) + cos ( x ) - cos ( x ) sin ( x )}{cos ( x ) sin ( x )}

\frac{cot ( x )}{cos ( x )} + \frac{1}{sin ( x )} - 1

Convert element to fraction:

1 = \frac{1}{1}

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

Least Common Multiplier of

cos (x), sin (x), 1 : cos (x) sin (x)

cos (x), sin (x), 1

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= cos (x) sin (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

cos (x) sin (x)

For

\frac{cot ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{cot ( x )}{cos ( x )} = \frac{cot ( x ) sin ( x )}{cos ( x ) sin ( x )}

For

\frac{1}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

\frac{1}{sin ( x )} = \frac{1 \cdot cos ( x )}{sin ( x ) cos ( x )} = \frac{cos ( x )}{cos ( x ) sin ( x )}

For

\frac{1}{1} :

multiply the denominator and numerator by

cos (x) sin (x)

\frac{1}{1} = \frac{1 \cdot cos ( x ) sin ( x )}{1 \cdot cos ( x ) sin ( x )} = \frac{cos ( x ) sin ( x )}{cos ( x ) sin ( x )}

= \frac{cot ( x ) sin ( x )}{cos ( x ) sin ( x )} + \frac{cos ( x )}{cos ( x ) sin ( x )} - \frac{cos ( x ) sin ( x )}{cos ( x ) sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{cot ( x ) sin ( x ) + cos ( x ) - cos ( x ) sin ( x )}{cos ( x ) sin ( x )}

\frac{cot ( x ) sin ( x ) + cos ( x ) - cos ( x ) sin ( x )}{cos ( x ) sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cot (x) sin (x) + cos (x) - cos (x) sin (x) = 0

Rewrite using trig identities

cos (x) - cos (x) sin (x) + cot (x) sin (x)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= cos (x) - cos (x) sin (x) + \frac{cos ( x )}{sin ( x )} sin (x)

Simplify

cos (x) - cos (x) sin (x) + \frac{cos ( x )}{sin ( x )} sin (x) : 2 cos (x) - cos (x) sin (x)

cos (x) - cos (x) sin (x) + \frac{cos ( x )}{sin ( x )} sin (x)

\frac{cos ( x )}{sin ( x )} sin (x) = cos (x)

\frac{cos ( x )}{sin ( x )} sin (x)

Multiply fractions:

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

= \frac{cos ( x ) sin ( x )}{sin ( x )}

Cancel the common factor:

sin (x)

= cos (x)

= cos (x) - cos (x) sin (x) + cos (x)

Add similar elements:

cos (x) + cos (x) = 2 cos (x)

= 2 cos (x) - cos (x) sin (x)

= 2 cos (x) - cos (x) sin (x)

2 cos (x) - cos (x) sin (x) = 0

Factor

2 cos (x) - cos (x) sin (x) : - cos (x) (sin (x) - 2)

2 cos (x) - cos (x) sin (x)

Factor out common term

- cos (x)

= - cos (x) (- 2 + sin (x))

- cos (x) (sin (x) - 2) = 0

Solving each part separately

cos (x) = 0 or sin (x) - 2 = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin (x) - 2 = 0 :

No Solution

sin (x) - 2 = 0

Move

2

to the right side

sin (x) - 2 = 0

Add

2

to both sides

sin (x) - 2 + 2 = 0 + 2

Simplify

sin (x) = 2

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (cot(x))/(cos(x))+1/(sin(x))=1 ?
The general solution for (cot(x))/(cos(x))+1/(sin(x))=1 is No Solution for x\in\mathbb{R}