Is 3csc(x)-cot(x)=(3-cos(x))/(sin(x)) ?

The answer to whether 3csc(x)-cot(x)=(3-cos(x))/(sin(x)) is True

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prove 3csc(x)-cot(x)=(3-cos(x))/(sin(x))

prove

3 csc (x) - cot (x) = \frac{3 - cos ( x )}{sin ( x )}

True

Solution steps

3 csc (x) - cot (x) = \frac{3 - cos ( x )}{sin ( x )}

Manipulating left side

3 csc (x) - cot (x)

Express with sin, cos

- cot (x) + 3 csc (x)

Use the basic trigonometric identity:

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

= - \frac{cos ( x )}{sin ( x )} + 3 csc (x)

Use the basic trigonometric identity:

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

= - \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{1}{sin ( x )}

Simplify

- \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{1}{sin ( x )} : \frac{- cos ( x ) + 3}{sin ( x )}

- \frac{cos ( x )}{sin ( x )} + 3 \cdot \frac{1}{sin ( x )}

3 \cdot \frac{1}{sin ( x )} = \frac{3}{sin ( x )}

3 \cdot \frac{1}{sin ( x )}

Multiply fractions:

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

= \frac{1 \cdot 3}{sin ( x )}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{sin ( x )}

= - \frac{cos ( x )}{sin ( x )} + \frac{3}{sin ( x )}

Apply rule

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

= \frac{- cos ( x ) + 3}{sin ( x )}

= \frac{- cos ( x ) + 3}{sin ( x )}

= \frac{3 - cos ( x )}{sin ( x )}

We showed that the two sides could take the same form

\Rightarrow True

Is 3csc(x)-cot(x)=(3-cos(x))/(sin(x)) ?
The answer to whether 3csc(x)-cot(x)=(3-cos(x))/(sin(x)) is True