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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{1 - tan ^{4} ( x )}{sec ^{2} ( x )}

Rewrite using trig identities

= - (tan (x) + 1) (tan (x) - 1)

Expand

- (- 1 + tan (x)) (1 + tan (x)) : - tan^{2} (x) + 1

= 1 - tan^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan^{2} (x) = sec (x) csc (x) tan (x) - 1

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

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

3 - sec^{2} (x) = 2 - tan^{2} (x)

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

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