Is (cos(x))/(1-sin^2(x))=sec(x) ?

The answer to whether (cos(x))/(1-sin^2(x))=sec(x) is True

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prove (cos(x))/(1-sin^2(x))=sec(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

1 - sin^{2} (x) = cos^{2} (x)

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

Cancel the common factor:

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 (cos(x))/(1-sin^2(x))=sec(x) ?
The answer to whether (cos(x))/(1-sin^2(x))=sec(x) is True