Is csc^3(x)= 1/(sin^3(x)) ?

The answer to whether csc^3(x)= 1/(sin^3(x)) is True

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prove csc^3(x)= 1/(sin^3(x))

prove

csc^{3} (x) = \frac{1}{sin ^{3} ( x )}

True

Solution steps

csc^{3} (x) = \frac{1}{sin ^{3} ( x )}

Manipulating left side

csc^{3} (x)

Express with sin, cos

csc^{3} (x)

Use the basic trigonometric identity:

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

= (\frac{1}{sin ( x )})^{3}

Simplify

(\frac{1}{sin ( x )})^{3} : \frac{1}{sin ^{3} ( x )}

(\frac{1}{sin ( x )})^{3}

Apply exponent rule:

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

= \frac{1 ^{3}}{sin ^{3} ( x )}

Apply rule

1^{a} = 1

1^{3} = 1

= \frac{1}{sin ^{3} ( x )}

= \frac{1}{sin ^{3} ( x )}

= \frac{1}{sin ^{3} ( x )}

We showed that the two sides could take the same form

\Rightarrow True

\frac{sec ( a )}{cot ( a ) + tan ( a )} = sin (a)

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

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

\frac{1}{2} \frac{tan ( b )}{1 - cos ^{2} ( b )} = csc (2 b)

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

Is csc^3(x)= 1/(sin^3(x)) ?
The answer to whether csc^3(x)= 1/(sin^3(x)) is True