Is sec(2x)=(1+tan^2(x))/(1-tan^2(x)) ?

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

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prove sec(2x)=(1+tan^2(x))/(1-tan^2(x))

prove

sec (2 x) = \frac{1 + tan ^{2} ( x )}{1 - tan ^{2} ( x )}

True

Solution steps

sec (2 x) = \frac{1 + tan ^{2} ( x )}{1 - tan ^{2} ( x )}

Manipulating right side

\frac{1 + tan ^{2} ( x )}{1 - tan ^{2} ( x )}

Express with sin, cos

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

Rewrite using trig identities

= \frac{1}{cos ( 2 x )}

Rewrite using trig identities

sec (2 x)

We showed that the two sides could take the same form

\Rightarrow True

cos (x) + tan (x) sin (x) = sec (x)

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

sec^{2} (θ) (1 - cos^{2} (θ)) = tan^{2} (θ)

cos^{2} (θ) + tan^{2} (θ) cos^{2} (θ) = 1

1 - cos (x) = sin (x)

Is sec(2x)=(1+tan^2(x))/(1-tan^2(x)) ?
The answer to whether sec(2x)=(1+tan^2(x))/(1-tan^2(x)) is True