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

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

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sin(x)cot(x)=1

sin (x) cot (x) = 1

No Solution for x \in R

Solution steps

sin (x) cot (x) = 1

Subtract

1

from both sides

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

Rewrite using trig identities

- 1 + cot (x) sin (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= cos (x)

= - 1 + cos (x)

- 1 + cos (x) = 0

Move

1

to the right side

- 1 + cos (x) = 0

Add

1

to both sides

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

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Since the equation is undefined for:

2 πn

No Solution for x \in R

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sin (x) - 1 - 3 sin^{2} (x) = 0

2 cos^{2} (x) + 2 cos (x) - 3 = 0

sec^{2} (x) cot (x) = 8, 0 < x < π

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