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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

cos (x)

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

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

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

Manipulating right side

cot (x) csc (x)

Express with sin, cos

cot (x) csc (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} : \frac{cos ( x )}{sin ^{2} ( x )}

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

Multiply fractions:

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

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

Multiply:

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

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

sin (2 t + π) = sin (2 t) cos (π) + cos (2 t) sin (π)

\frac{tan ( x ) - 1}{sec ( x )} = sin (x) - cos (x)

cot^{2} (α) = cos^{2} (α) + (cot (α) \cdot cos (α))^{2}

\frac{1}{tan ( α )} \cdot \frac{1}{cos ( α )} = \frac{1}{sin ( α )}

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