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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Factor

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

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

Rewrite using trig identities

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

Expand

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

= 1 - tan^{4} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (π) = sin (- π)

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

csc (t) cot (t) = \frac{1 + cot ^{2} ( t )}{sec ( t )}

csc (a) sec (a) = 2 csc (2 a)

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

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