What is the general solution for tan(x)csc(x)-tan(x)=0 ?

The general solution for tan(x)csc(x)-tan(x)=0 is No Solution for x\in\mathbb{R}

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

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

No Solution for x \in R

Solution steps

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

Factor

tan (x) csc (x) - tan (x) : tan (x) (csc (x) - 1)

tan (x) csc (x) - tan (x)

Factor out common term

tan (x)

= tan (x) (csc (x) - 1)

tan (x) (csc (x) - 1) = 0

Solving each part separately

tan (x) = 0 or csc (x) - 1 = 0

tan (x) = 0 : x = πn

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

csc (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

csc (x) - 1 = 0

Move

1

to the right side

csc (x) - 1 = 0

Add

1

to both sides

csc (x) - 1 + 1 = 0 + 1

Simplify

csc (x) = 1

csc (x) = 1

General solutions for

csc (x) = 1

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

x = πn, x = \frac{π}{2} + 2 πn

Since the equation is undefined for:

πn, \frac{π}{2} + 2 πn

No Solution for x \in R

cos (6 0^{\circ} - x) = sin (70 - 3 x)

sin (- x) = sin (x)

sin (x) = a

sin (6 x) - sin (2 x) = 0

z, cos (x) + cos (y) + cos (z) = 0

What is the general solution for tan(x)csc(x)-tan(x)=0 ?
The general solution for tan(x)csc(x)-tan(x)=0 is No Solution for x\in\mathbb{R}