Is sin(x)cos(x)csc^2(x)=cot(x) ?

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

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prove sin(x)cos(x)csc^2(x)=cot(x)

prove

sin (x) cos (x) csc^{2} (x) = cot (x)

True

Solution steps

sin (x) cos (x) csc^{2} (x) = cot (x)

Manipulating left side

sin (x) cos (x) csc^{2} (x)

Express with sin, cos

cos (x) csc^{2} (x) sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

(\frac{1}{sin ( x )})^{2} = \frac{1}{sin ^{2} ( x )}

(\frac{1}{sin ( x )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( x )}

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

sin (x)

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

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

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

Use the basic trigonometric identity:

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

= cot (x)

We showed that the two sides could take the same form

\Rightarrow True

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