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

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

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

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

No Solution for x \in R

Solution steps

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

Subtract

cos (x)

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

tan (θ) = - \frac{3}{7}

sin (2 x) = 0, - π \leq x \leq π

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

2 cos (θ) = 3 cos (3 θ)

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

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