Is (cot(θ)sec(θ))/(csc(θ))=1 ?

The answer to whether (cot(θ)sec(θ))/(csc(θ))=1 is True

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prove (cot(θ)sec(θ))/(csc(θ))=1

prove

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

True

Solution steps

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

Manipulating left side

\frac{cot ( θ ) sec ( θ )}{csc ( θ )}

Express with sin, cos

\frac{cot ( θ ) sec ( θ )}{csc ( θ )}

Use the basic trigonometric identity:

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

= \frac{\frac{c o s ( θ )}{s i n ( θ )} sec ( θ )}{csc ( θ )}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

Apply the fraction rule:

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

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

Apply the fraction rule:

\frac{a}{1} = a

= \frac{cos ( θ )}{sin ( θ )} \cdot \frac{1}{cos ( θ )} sin (θ)

Multiply fractions:

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

= \frac{cos ( θ ) \cdot 1 \cdot sin ( θ )}{sin ( θ ) cos ( θ )}

Cancel the common factor:

cos (θ)

= \frac{1 \cdot sin ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

Is (cot(θ)sec(θ))/(csc(θ))=1 ?
The answer to whether (cot(θ)sec(θ))/(csc(θ))=1 is True