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

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

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prove 1-tan^4(A)=2sec^2(A)-sec^4(A)

prove

1 - tan^{4} (A) = 2 sec^{2} (A) - sec^{4} (A)

True

Solution steps

1 - tan^{4} (A) = 2 sec^{2} (A) - sec^{4} (A)

Manipulating right side

2 sec^{2} (A) - sec^{4} (A)

Factor

- sec^{4} (A) + 2 sec^{2} (A) : sec^{2} (A) (- sec^{2} (A) + 2)

= (2 - sec^{2} (A)) sec^{2} (A)

Rewrite using trig identities

= (- tan^{2} (A) + 1) (tan^{2} (A) + 1)

Expand

(1 + tan^{2} (A)) (1 - tan^{2} (A)) : 1 - tan^{4} (A)

= 1 - tan^{4} (A)

We showed that the two sides could take the same form

\Rightarrow True

cos^{4} (θ) - sin^{4} (θ) = 2 cos^{2} (θ) - 1

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

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

cos (4 θ) = 8 cos^{4} (θ) - 8 cos^{2} (θ) + 1

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

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