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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec^{4} (x)

Express with sin, cos

sec^{4} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{4} = 1

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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