Is (sec(x))/(tan(x))=csc(x) ?

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

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prove (sec(x))/(tan(x))=csc(x)

prove

\frac{sec ( x )}{tan ( x )} = csc (x)

True

Solution steps

\frac{sec ( x )}{tan ( x )} = csc (x)

Manipulating left side

\frac{sec ( x )}{tan ( x )}

Express with sin, cos

\frac{sec ( x )}{tan ( x )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Refine

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

Cancel the common factor:

cos (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( x )}}

Simplify

\frac{1}{\frac{1}{c s c ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1 - sin ( v )}{cos ( v )} + \frac{cos ( v )}{1 - sin ( v )} = 2 sec (v)

sec (θ) - tan (θ) = \frac{1}{sec ( θ ) + tan ( θ )}

sec (x) = \frac{1 + csc ( x )}{cot ( x ) + cos ( x )}

9 sec^{2} (θ) - 5 tan^{2} (θ) = 5 + 4 sec^{2} (θ)

cot (2 θ) = cot (θ) - \frac{1}{2} sec (θ) csc (θ)

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