What is the general solution for 1/(cot(x))-(sec(x))/(csc(x))=cos(x) ?

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

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Popular Trigonometry >

1/(cot(x))-(sec(x))/(csc(x))=cos(x)

Solution

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} = cos (x)

Solution

No Solution for x \in R

Solution steps

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} = cos (x)

Subtract

cos (x)

from both sides

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x) = 0

Simplify

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x) : \frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

\frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - cos (x)

Convert element to fraction:

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

= \frac{1}{cot ( x )} - \frac{sec ( x )}{csc ( x )} - \frac{cos ( x )}{1}

Least Common Multiplier of

cot (x), csc (x), 1 : cot (x) csc (x)

cot (x), csc (x), 1

Lowest Common Multiplier (LCM)

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

= cot (x) csc (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

cot (x) csc (x)

For

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

multiply the denominator and numerator by

csc (x)

\frac{1}{cot ( x )} = \frac{1 \cdot csc ( x )}{cot ( x ) csc ( x )} = \frac{csc ( x )}{cot ( x ) csc ( x )}

For

\frac{sec ( x )}{csc ( x )} :

multiply the denominator and numerator by

cot (x)

\frac{sec ( x )}{csc ( x )} = \frac{sec ( x ) cot ( x )}{csc ( x ) cot ( x )}

For

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

multiply the denominator and numerator by

cot (x) csc (x)

\frac{cos ( x )}{1} = \frac{cos ( x ) cot ( x ) csc ( x )}{1 \cdot cot ( x ) csc ( x )} = \frac{cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

= \frac{csc ( x )}{cot ( x ) csc ( x )} - \frac{sec ( x ) cot ( x )}{csc ( x ) cot ( x )} - \frac{cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )}

\frac{csc ( x ) - sec ( x ) cot ( x ) - cos ( x ) cot ( x ) csc ( x )}{cot ( x ) csc ( x )} = 0

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

csc (x) - sec (x) cot (x) - cos (x) cot (x) csc (x) = 0

Express with sin, cos

csc (x) - cot (x) sec (x) - cos (x) cot (x) csc (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

\frac{1}{sin ( x )} - \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} - cos (x) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} : - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

cos (x) \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} = \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

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

Multiply fractions:

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

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

cos (x) \cdot 1 \cdot cos (x) = cos^{2} (x)

cos (x) \cdot 1 \cdot cos (x)

Apply exponent rule:

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

cos (x) cos (x) = cos^{1 + 1} (x)

= 1 \cdot cos^{1 + 1} (x)

Refine

= cos^{2} (x)

= \frac{cos ^{2} ( x )}{sin ( x ) sin ( x )}

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

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

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

= \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1}{sin ( x )} - \frac{1}{sin ( x )} - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

Add similar elements:

1 \cdot \frac{1}{sin ( x )} - 1 \cdot \frac{1}{sin ( x )} = 0

= - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= - \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

- \frac{cos ^{2} ( x )}{sin ^{2} ( x )} = 0

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

- cos^{2} (x) = 0

Divide both sides by

- 1

- cos^{2} (x) = 0

Divide both sides by

- 1

- cos^{2} (x) = 0

Divide both sides by

- 1

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

Simplify

cos^{2} (x) = 0

cos^{2} (x) = 0

Apply rule

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

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

Since the equation is undefined for:

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

No Solution for x \in R

Graph

Popular Examples

sin(x)=1,2pi<= x<= 4pi

sin (x) = 1, 2 π \leq x \leq 4 π

cos(a)=1

cos (a) = 1

-cos(x)+1=2sin^2(x)

- cos (x) + 1 = 2 sin^{2} (x)

sec(θ)= 7/5

sec (θ) = \frac{7}{5}

(sin(x))/(cos(x))=0

\frac{sin ( x )}{cos ( x )} = 0

Frequently Asked Questions (FAQ)

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