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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

= cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

sec (b) cot (b) = csc (b) tan (b)

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

cos (π θ) = - cos (θ)

csc (x) tan (x) sec (x) = sec^{2} (x)

tan (x) sec^{4} (x) = \frac{sin ( x )}{cos ^{5} ( x )}

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