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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

Multiply

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

= 1

= \frac{1}{1}

Apply rule

\frac{a}{1} = a

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

sec (x) = sin (x) \cdot (tan (x) + cot (x))

(cot (θ) + tan (θ)) sin (θ) = sec (θ)

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

cos (A) = 2 cos (A) sin (B)

cos (18 0^{\circ} - θ) = - cos (θ)

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