Is tan^3(x)(csc^3(x))=sec^3(x) ?

The answer to whether tan^3(x)(csc^3(x))=sec^3(x) is True

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prove tan^3(x)(csc^3(x))=sec^3(x)

prove

tan^{3} (x) (csc^{3} (x)) = sec^{3} (x)

True

Solution steps

tan^{3} (x) csc^{3} (x) = sec^{3} (x)

Manipulating left side

tan^{3} (x) csc^{3} (x)

Express with sin, cos

csc^{3} (x) tan^{3} (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

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

Apply exponent rule:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin^{3} (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

\frac{1}{( \frac{1}{s e c ( x )} ) ^{3}}

Simplify

\frac{1}{( \frac{1}{s e c ( x )} ) ^{3}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{3} = 1

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

= \frac{1}{\frac{1}{s e c ^{3} ( x )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

Apply rule

\frac{a}{1} = a

= sec^{3} (x)

sec^{3} (x)

sec^{3} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (x) sec (x) = sin (x)

tan^{2} (y) - sin^{2} (y) = tan^{2} (y) sin^{2} (y)

sin (θ) (1 + cot^{2} (θ)) = csc (θ)

csc (x) - sin (x) = csc (x) cot (x)

5 + cot^{2} (x) = 4 + csc^{2} (x)

Is tan^3(x)(csc^3(x))=sec^3(x) ?
The answer to whether tan^3(x)(csc^3(x))=sec^3(x) is True