Is cot(-x)sin(-x)=cos(x) ?

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

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prove cot(-x)sin(-x)=cos(x)

prove

cot (- x) sin (- x) = cos (x)

True

Solution steps

cot (- x) sin (- x) = cos (x)

Manipulating left side

cot (- x) sin (- x)

Use the negative angle identity:

sin (- x) = - sin (x)

= cot (- x) (- sin (x))

Use the negative angle identity:

cot (- x) = - cot (x)

= (- cot (x)) (- sin (x))

Express with sin, cos

(- cot (x)) (- sin (x))

Use the basic trigonometric identity:

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

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

Simplify

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

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

Remove parentheses:

(- a) = - a, - (- a) = a

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= cos (x)

= cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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