Is (1+sin(x))/(sin(x))=1+csc(x) ?

The answer to whether (1+sin(x))/(sin(x))=1+csc(x) is True

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prove (1+sin(x))/(sin(x))=1+csc(x)

prove

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

True

Solution steps

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

Manipulating right side

1 + csc (x)

Express with sin, cos

1 + csc (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

Is (1+sin(x))/(sin(x))=1+csc(x) ?
The answer to whether (1+sin(x))/(sin(x))=1+csc(x) is True