What is the general solution for cos(x)=(tan(x)+cot(x))/(csc(x)) ?

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

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Popular >

cos(x)=(tan(x)+cot(x))/(csc(x))

Solution

cos (x) = \frac{tan ( x ) + cot ( x )}{csc ( x )}

Solution

No Solution for x \in R

Solution steps

cos (x) = \frac{tan ( x ) + cot ( x )}{csc ( x )}

Subtract

\frac{tan ( x ) + cot ( x )}{csc ( x )}

from both sides

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

Simplify

cos (x) - \frac{tan ( x ) + cot ( x )}{csc ( x )} : \frac{cos ( x ) csc ( x ) - tan ( x ) - cot ( x )}{csc ( x )}

cos (x) - \frac{tan ( x ) + cot ( x )}{csc ( x )}

Convert element to fraction:

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

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

- (tan (x) + cot (x)) : - tan (x) - cot (x)

- (tan (x) + cot (x))

Distribute parentheses

= - (tan (x)) - (cot (x))

Apply minus-plus rules

+ (- a) = - a

= - tan (x) - cot (x)

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

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

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

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

Rewrite using trig identities

- cot (x) - tan (x) + cos (x) csc (x)

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Express with sin, cos

csc (x) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

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

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

= - cot (x) - tan (x) + cot (x)

Simplify

= - tan (x)

- tan (x) = 0

Divide both sides by

- 1

- tan (x) = 0

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{0}{- 1}

Simplify

tan (x) = 0

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

Since the equation is undefined for:

πn

No Solution for x \in R

Popular Examples

-tan^2(x)-1=0

- tan^{2} (x) - 1 = 0

2cot^2(y)+3csc(y)=0

2 cot^{2} (y) + 3 csc (y) = 0

cot(x)=-0.456

cot (x) = - 0.456

4tan^2(θ)-2tan(θ)+3=-7tan(θ)+2

4 tan^{2} (θ) - 2 tan (θ) + 3 = - 7 tan (θ) + 2

-2=csc(θ)

- 2 = csc (θ)

Frequently Asked Questions (FAQ)

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