What is the general solution for (1+sin(x))/(sin(x))=1+cos(x) ?

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

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(1+sin(x))/(sin(x))=1+cos(x)

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

No Solution for x \in R

Solution steps

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

Subtract

1 + cos (x)

from both sides

\frac{1 + sin ( x )}{sin ( x )} - 1 - cos (x) = 0

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

sin (x) - 1 \cdot sin (x) = 0

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

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

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

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

Rewrite using trig identities

1 - cos (x) sin (x)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

= 1 - \frac{sin ( 2 x )}{2}

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

- \frac{sin ( 2 x )}{2} = - 1

- \frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

- \frac{sin ( 2 x )}{2} = - 1

Multiply both sides by

2

2 (- \frac{sin ( 2 x )}{2}) = 2 (- 1)

Simplify

- sin (2 x) = - 2

- sin (2 x) = - 2

Divide both sides by

- 1

- sin (2 x) = - 2

Divide both sides by

- 1

\frac{- sin ( 2 x )}{- 1} = \frac{- 2}{- 1}

Simplify

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

No Solution for x \in R

cos (3 x) - cos (5 x) = 0

2 = 2 cos (x)

4 cos (x) = 4 cos (2 x)

cos (θ) = 513, 3 π^{2} < θ < 2 π

cos (x) + ∣ cos (x) ∣ = 1

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