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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec^{2} (B)

Express with sin, cos

sec^{2} (B)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

3 sin^{2} (2 x) = 12 cos^{2} (x) sin^{2} (x)

cos (θ) = sin (θ) cot (θ)

sin (θ) cos (θ) = 1

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

csc (θ) - csc (θ) cos^{2} (θ) = sin (θ)

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