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

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

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

prove

tan (- x) cos (x) = - sin (x)

True

Solution steps

tan (- x) cos (x) = - sin (x)

Manipulating left side

tan (- x) cos (x)

Use the negative angle identity:

tan (- x) = - tan (x)

= cos (x) (- tan (x))

Express with sin, cos

(- tan (x)) cos (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= - sin (x)

= - sin (x)

= - sin (x)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e tan (2 π - x) = - tan (x)

p ro v e sec (2 x) = \frac{sec ^{2} ( x )}{2 - sec ^{2} ( x )}

p ro v e tan (a) + cot (a) = sec (a) csc (a)

p ro v e (1 - sin (x)) (1 + sin (x)) = cos^{2} (x)

p ro v e (csc^{2} (x) - 1) sin (x) = cos^{2} (x) csc (x)