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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Rewrite using trig identities

= cos (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

cot (θ) + \frac{sin ( θ )}{1 + cos ( θ )} = csc (θ)

\frac{sin ( \frac{π}{2} + x )}{sin ( π - x )} = cot (x)

cos (x) \cdot csc (x) = cot (x)

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