Is 1/(cos(x)csc(x))=tan(x) ?

The answer to whether 1/(cos(x)csc(x))=tan(x) is True

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prove 1/(cos(x)csc(x))=tan(x)

prove

\frac{1}{cos ( x ) csc ( x )} = tan (x)

True

Solution steps

\frac{1}{cos ( x ) csc ( x )} = tan (x)

Manipulating left side

\frac{1}{cos ( x ) csc ( x )}

Express with sin, cos

\frac{1}{cos ( x ) csc ( x )}

Use the basic trigonometric identity:

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

= \frac{1}{cos ( x ) \frac{1}{s i n ( x )}}

Simplify

\frac{1}{cos ( x ) \frac{1}{s i n ( x )}} : \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x ) \frac{1}{s i n ( x )}}

Multiply

cos (x) \frac{1}{sin ( x )} : \frac{cos ( x )}{sin ( x )}

cos (x) \frac{1}{sin ( x )}

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

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

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

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

= \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

\frac{1}{cos ^{2} ( x )} = tan^{2} (x) + 1

1 - cos (2 x) = 2 sin^{2} (x)

\frac{sin ^{2} ( t )}{tan ^{2} ( t )} = cos^{2} (t)

tan (- θ) = - tan (θ)

\frac{tan ( x )}{1 - cos ^{2} ( x )} = sec (x) csc (x)

Is 1/(cos(x)csc(x))=tan(x) ?
The answer to whether 1/(cos(x)csc(x))=tan(x) is True