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

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

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prove sec^2(a)= 1/(1-sin^2(a))

prove

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

True

Solution steps

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

Manipulating left side

sec^{2} (a)

Express with sin, cos

sec^{2} (a)

Use the basic trigonometric identity:

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

= (\frac{1}{cos ( a )})^{2}

Simplify

(\frac{1}{cos ( a )})^{2} : \frac{1}{cos ^{2} ( a )}

(\frac{1}{cos ( a )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cos ^{2} ( a )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( a )}

= \frac{1}{cos ^{2} ( a )}

= \frac{1}{cos ^{2} ( a )}

Rewrite using trig identities

\frac{1}{cos ^{2} ( a )}

Use the Pythagorean identity:

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

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

= \frac{1}{1 - sin ^{2} ( a )}

= \frac{1}{1 - sin ^{2} ( a )}

We showed that the two sides could take the same form

\Rightarrow True

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