Is (cot(x)sec(x))/(csc(x))=1 ?

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

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prove (cot(x)sec(x))/(csc(x))=1

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

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

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

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

cos (x)

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

Cancel the common factor:

sin (x)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

1 + cot^{2} (θ) = csc^{2} (θ)

(csc^{2} (x) - 1) sin (x) = cos^{2} (x) csc (x)

\frac{1 + cos ( 2 x )}{sin ( 2 x )} = cot (x)

sin (x) cot (x) = cos (x)

sin (π + x) = - sin (x)

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