Is (csc(a))/(cot(a))=sec(a) ?

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

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prove (csc(a))/(cot(a))=sec(a)

prove

\frac{csc ( a )}{cot ( a )} = sec (a)

True

Solution steps

\frac{csc ( a )}{cot ( a )} = sec (a)

Manipulating left side

\frac{csc ( a )}{cot ( a )}

Express with sin, cos

\frac{csc ( a )}{cot ( a )}

Use the basic trigonometric identity:

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

= \frac{\frac{1}{s i n ( a )}}{cot ( a )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

sin (a)

= \frac{1}{cos ( a )}

= \frac{1}{cos ( a )}

= \frac{1}{cos ( a )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( a )}}

Simplify

\frac{1}{\frac{1}{s e c ( a )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ( a )}{1}

Apply rule

\frac{a}{1} = a

= sec (a)

sec (a)

sec (a)

We showed that the two sides could take the same form

\Rightarrow True

(tan (x) - 1) cos (x) = sin (x) - cos (x)

sin (θ - \frac{π}{2}) = - cos (θ)

sec (x) - cos (x) = tan (x) \cdot sin (x)

\frac{cot ( x ) - 1}{1 - tan ( x )} = cot (x)

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

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