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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec (2 x)

Express with sin, cos

sec (2 x)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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sin (x) tan (x) = cos (x)

2 sin^{2} (\frac{x}{2}) + 4 cos^{2} (\frac{x}{2}) - 3 = cos (x)

tan (2 u) = \frac{( 2 tan ( u ) )}{1 - tan ^{2} ( u )}

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