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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Simplify

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Cancel the common factor:

sin (x)

= 1

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

tan (\frac{π}{2} - u) = cot (u)

csc (θ) - cot (θ) = \frac{sin ( θ )}{1 + cos ( θ )}

csc^{2} (x) sec (x) = sec (x) + csc (x) cot (x)

1 - cot^{4} (x) = 2 csc^{2} (x) - csc^{4} (x)

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