Is (cot(x))/(csc(x))=cos(x) ?

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

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prove (cot(x))/(csc(x))=cos(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

sin (x)

= cos (x)

= cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

1 + tan^{2} (α) = \frac{1}{cos ^{2} ( α )}

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

\frac{tan ( x ) - cot ( x )}{tan ( x ) + cot ( x )} = 1 - 2 cos^{2} (x)

tan (θ) cos (θ) = sin (θ)

\frac{sin ( x + \frac{π}{2} )}{sin ( π - x )} = cot (x)

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