Is 1/(cot(x)cos(x)tan(x))=sec(x) ?

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

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prove 1/(cot(x)cos(x)tan(x))=sec(x)

prove

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

True

Solution steps

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

Manipulating left side

\frac{1}{cot ( x ) cos ( x ) tan ( x )}

Express with sin, cos

\frac{1}{cos ( x ) cot ( x ) tan ( x )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Cancel the common factor:

sin (x)

= cos (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ^{2} ( a )}{1 - sin ( a )} = 1 + sin (a)

2 sin (- t) cos (- t) = 2 sin (- t) cos (t)

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

csc (θ) \cdot tan (θ) = sec (θ)

cot (θ) \cdot csc (θ) \cdot sec (θ) = csc^{2} (θ)

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