Is sin(x)*(1+cot(x))=sin(x)+cos(x) ?

The answer to whether sin(x)*(1+cot(x))=sin(x)+cos(x) is True

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prove sin(x)*(1+cot(x))=sin(x)+cos(x)

prove

sin (x) \cdot (1 + cot (x)) = sin (x) + cos (x)

True

Solution steps

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

Manipulating left side

sin (x) (1 + cot (x))

Express with sin, cos

(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)

Simplify

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

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

Join

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= sin (x) + cos (x)

= sin (x) + cos (x)

= cos (x) + sin (x)

= sin (x) + cos (x)

We showed that the two sides could take the same form

\Rightarrow True

Is sin(x)*(1+cot(x))=sin(x)+cos(x) ?
The answer to whether sin(x)*(1+cot(x))=sin(x)+cos(x) is True