Is (cos(x)csc(x))/(cot^2(x))=tan(x) ?

The answer to whether (cos(x)csc(x))/(cot^2(x))=tan(x) is True

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prove (cos(x)csc(x))/(cot^2(x))=tan(x)

prove

\frac{cos ( x ) csc ( x )}{cot ^{2} ( x )} = tan (x)

True

Solution steps

\frac{cos ( x ) csc ( x )}{cot ^{2} ( x )} = tan (x)

Manipulating left side

\frac{cos ( x ) csc ( x )}{cot ^{2} ( x )}

Express with sin, cos

\frac{cos ( x ) csc ( x )}{cot ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{cos ( x ) \frac{1}{s i n ( x )}}{cot ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{cos ( x ) \frac{1}{s i n ( x )}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

Simplify

\frac{cos ( x ) \frac{1}{s i n ( x )}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}} : \frac{sin ( x )}{cos ( x )}

\frac{cos ( x ) \frac{1}{s i n ( x )}}{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}

(\frac{cos ( x )}{sin ( x )})^{2} = \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

(\frac{cos ( x )}{sin ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= \frac{\frac{1}{s i n ( x )} cos ( x )}{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

Multiply

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

= \frac{\frac{c o s ( x )}{s i n ( x )}}{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}

Divide fractions:

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

= \frac{cos ( x ) sin ^{2} ( x )}{sin ( x ) cos ^{2} ( x )}

Cancel the common factor:

cos (x)

= \frac{sin ^{2} ( x )}{sin ( x ) cos ( x )}

Cancel the common factor:

sin (x)

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

Is (cos(x)csc(x))/(cot^2(x))=tan(x) ?
The answer to whether (cos(x)csc(x))/(cot^2(x))=tan(x) is True