Is (-csc(x))/(sec(x))+(cos(x))/(-sin(x))=-2cot(x) ?

The answer to whether (-csc(x))/(sec(x))+(cos(x))/(-sin(x))=-2cot(x) is True

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prove (-csc(x))/(sec(x))+(cos(x))/(-sin(x))=-2cot(x)

prove

\frac{- csc ( x )}{sec ( x )} + \frac{cos ( x )}{- sin ( x )} = - 2 cot (x)

True

Solution steps

\frac{- csc ( x )}{sec ( x )} + \frac{cos ( x )}{- sin ( x )} = - 2 cot (x)

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{- \frac{1}{s i n ( x )}}{sec ( x )} + \frac{cos ( x )}{- sin ( x )}

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{- \frac{1}{s i n ( x )}}{\frac{1}{c o s ( x )}} + \frac{cos ( x )}{- sin ( x )}

Simplify

\frac{- \frac{1}{s i n ( x )}}{\frac{1}{c o s ( x )}} + \frac{cos ( x )}{- sin ( x )} : - \frac{2 cos ( x )}{sin ( x )}

\frac{- \frac{1}{s i n ( x )}}{\frac{1}{c o s ( x )}} + \frac{cos ( x )}{- sin ( x )}

\frac{- \frac{1}{s i n ( x )}}{\frac{1}{c o s ( x )}} = - \frac{cos ( x )}{sin ( x )}

\frac{- \frac{1}{s i n ( x )}}{\frac{1}{c o s ( x )}}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{1}{s i n ( x )}}{\frac{1}{c o s ( x )}}

Divide fractions:

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

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

Refine

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

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

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

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

Apply rule

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

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

Add similar elements:

- cos (x) - cos (x) = - 2 cos (x)

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

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

- 2 cot (x)

- 2 cot (x)

We showed that the two sides could take the same form

\Rightarrow True

Is (-csc(x))/(sec(x))+(cos(x))/(-sin(x))=-2cot(x) ?
The answer to whether (-csc(x))/(sec(x))+(cos(x))/(-sin(x))=-2cot(x) is True