Is (cos(pi/2+x))/(cos(pi+x))=tan(x) ?

The answer to whether (cos(pi/2+x))/(cos(pi+x))=tan(x) is True

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prove (cos(pi/2+x))/(cos(pi+x))=tan(x)

prove

\frac{cos ( \frac{π}{2} + x )}{cos ( π + x )} = tan (x)

True

Solution steps

\frac{cos ( \frac{π}{2} + x )}{cos ( π + x )} = tan (x)

Manipulating left side

\frac{cos ( \frac{π}{2} + x )}{cos ( π + x )}

Rewrite using trig identities

= \frac{- sin ( x )}{cos ( π + x )}

Rewrite using trig identities

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

Apply the fraction rule:

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

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

Use the basic trigonometric identity:

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

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

cot (x) sec (x) = csc (x)

1 - cos (2 x) = tan (x) sin (2 x)

1 = sec^{2} (2 x) - tan^{2} (2 x)

\frac{sin ^{2} ( y )}{cos ( y )} = sec (y) - cos (y)

cot (θ) sec (θ) = csc (θ)

Is (cos(pi/2+x))/(cos(pi+x))=tan(x) ?
The answer to whether (cos(pi/2+x))/(cos(pi+x))=tan(x) is True