Is 1/(sec^3(x)cos^4(x))=sec(x) ?

The answer to whether 1/(sec^3(x)cos^4(x))=sec(x) is True

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prove 1/(sec^3(x)cos^4(x))=sec(x)

prove

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )} = sec (x)

True

Solution steps

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )} = sec (x)

Manipulating left side

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )}

Express with sin, cos

\frac{1}{cos ^{4} ( x ) sec ^{3} ( x )}

Use the basic trigonometric identity:

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

= \frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}}

Simplify

\frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}} : \frac{1}{cos ( x )}

\frac{1}{cos ^{4} ( x ) ( \frac{1}{c o s ( x )} ) ^{3}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{3} = 1

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

= \frac{1}{\frac{1}{c o s ^{3} ( x )} cos ^{4} ( x )}

Multiply

cos^{4} (x) \frac{1}{cos ^{3} ( x )} : cos (x)

cos^{4} (x) \frac{1}{cos ^{3} ( x )}

Multiply fractions:

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

= \frac{1 \cdot cos ^{4} ( x )}{cos ^{3} ( x )}

Multiply:

1 \cdot cos^{4} (x) = cos^{4} (x)

= \frac{cos ^{4} ( x )}{cos ^{3} ( x )}

Apply exponent rule:

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

\frac{cos ^{4} ( x )}{cos ^{3} ( x )} = cos^{4 - 3} (x)

= cos^{4 - 3} (x)

Subtract the numbers:

4 - 3 = 1

= cos (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

We showed that the two sides could take the same form

\Rightarrow True

Is 1/(sec^3(x)cos^4(x))=sec(x) ?
The answer to whether 1/(sec^3(x)cos^4(x))=sec(x) is True