Is cos(x)=sin(x)tan^2(x)cot^3(x) ?

The answer to whether cos(x)=sin(x)tan^2(x)cot^3(x) is True

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prove cos(x)=sin(x)tan^2(x)cot^3(x)

prove

cos (x) = sin (x) tan^{2} (x) cot^{3} (x)

True

Solution steps

cos (x) = sin (x) tan^{2} (x) cot^{3} (x)

Manipulating right side

sin (x) tan^{2} (x) cot^{3} (x)

Express with sin, cos

cot^{3} (x) sin (x) tan^{2} (x)

Use the basic trigonometric identity:

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

= (\frac{cos ( x )}{sin ( x )})^{3} sin (x) tan^{2} (x)

Use the basic trigonometric identity:

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

= (\frac{cos ( x )}{sin ( x )})^{3} sin (x) (\frac{sin ( x )}{cos ( x )})^{2}

Simplify

(\frac{cos ( x )}{sin ( x )})^{3} sin (x) (\frac{sin ( x )}{cos ( x )})^{2} : cos (x)

(\frac{cos ( x )}{sin ( x )})^{3} sin (x) (\frac{sin ( x )}{cos ( x )})^{2}

(\frac{cos ( x )}{sin ( x )})^{3} = \frac{cos ^{3} ( x )}{sin ^{3} ( x )}

(\frac{cos ( x )}{sin ( x )})^{3}

Apply exponent rule:

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

= \frac{cos ^{3} ( x )}{sin ^{3} ( x )}

= (\frac{sin ( x )}{cos ( x )})^{2} \frac{cos ^{3} ( x )}{sin ^{3} ( x )} sin (x)

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

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

Apply exponent rule:

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

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

= \frac{cos ^{3} ( x )}{sin ^{3} ( x )} \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )} sin (x)

Multiply fractions:

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

= \frac{cos ^{3} ( x ) sin ^{2} ( x ) sin ( x )}{sin ^{3} ( x ) cos ^{2} ( x )}

cos^{3} (x) sin^{2} (x) sin (x) = cos^{3} (x) sin^{3} (x)

cos^{3} (x) sin^{2} (x) sin (x)

Apply exponent rule:

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

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= cos^{3} (x) sin^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= cos^{3} (x) sin^{3} (x)

= \frac{cos ^{3} ( x ) sin ^{3} ( x )}{sin ^{3} ( x ) cos ^{2} ( x )}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{cos ^{3} ( x )}{cos ^{2} ( x )} = cos^{3 - 2} (x)

= \frac{sin ^{3} ( x ) cos ^{3 - 2} ( x )}{sin ^{3} ( x )}

Subtract the numbers:

3 - 2 = 1

= \frac{sin ^{3} ( x ) cos ( x )}{sin ^{3} ( x )}

Cancel the common factor:

sin^{3} (x)

= cos (x)

= cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

Is cos(x)=sin(x)tan^2(x)cot^3(x) ?
The answer to whether cos(x)=sin(x)tan^2(x)cot^3(x) is True