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

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

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sin(x)tan(x)= 1/(cot(x))

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

No Solution for x \in R

Solution steps

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

Subtract

\frac{1}{cot ( x )}

from both sides

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

Simplify

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

sin (x) tan (x) - \frac{1}{cot ( x )}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

sin (x) tan (x) cot (x) - 1 = 0

Rewrite using trig identities

- 1 + cot (x) sin (x) tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cot (x)

= 1 \cdot sin (x)

Multiply:

1 \cdot sin (x) = sin (x)

= sin (x)

= - 1 + sin (x)

- 1 + sin (x) = 0

Move

1

to the right side

- 1 + sin (x) = 0

Add

1

to both sides

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

Simplify

sin (x) = 1

sin (x) = 1

General solutions for

sin (x) = 1

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

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution for x \in R

0 = - 2 sin (x) - 1

x, z = sin (2 x + 3 y)

2 sin (3 θ) = 2

2 tan (x) csc (x) + 2 csc (x) + tan (x) = - 1

4 cos (x) + 1 = 2 cos (x)

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