Is (cos^4(x)-sin^4(x))/(cos^2(x))=1-tan^2(x) ?

The answer to whether (cos^4(x)-sin^4(x))/(cos^2(x))=1-tan^2(x) is True

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prove (cos^4(x)-sin^4(x))/(cos^2(x))=1-tan^2(x)

prove

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

True

Solution steps

\frac{cos ^{4} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )} = 1 - tan^{2} (x)

Manipulating left side

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

Rewrite using trig identities

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

Manipulating right side

1 - tan^{2} (x)

Rewrite using trig identities

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

Express with sin, cos

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

Rewrite using trig identities

= \frac{cos ^{4} ( x ) - sin ^{4} ( x )}{1 - sin ^{2} ( x )}

We showed that the two sides could take the same form

\Rightarrow True

csc^{2} (x) + cot^{2} (x) = 2 csc^{2} (x) - 1

cot (\frac{π}{2} - x) = tan (x)

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

csc^{2} (x) = 1 + cot^{2} (x)

cot^{2} (x) - csc^{2} (x) = - 1

Is (cos^4(x)-sin^4(x))/(cos^2(x))=1-tan^2(x) ?
The answer to whether (cos^4(x)-sin^4(x))/(cos^2(x))=1-tan^2(x) is True