Is tan^4(t)+tan^2(t)=sec^4(t)-sec^2(t) ?

The answer to whether tan^4(t)+tan^2(t)=sec^4(t)-sec^2(t) is True

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prove tan^4(t)+tan^2(t)=sec^4(t)-sec^2(t)

prove

tan^{4} (t) + tan^{2} (t) = sec^{4} (t) - sec^{2} (t)

True

Solution steps

tan^{4} (t) + tan^{2} (t) = sec^{4} (t) - sec^{2} (t)

Manipulating left side

tan^{4} (t) + tan^{2} (t)

Factor

tan^{2} (t) + tan^{4} (t) : tan^{2} (t) (1 + tan^{2} (t))

= (1 + tan^{2} (t)) tan^{2} (t)

Rewrite using trig identities

= sec^{2} (t) (sec^{2} (t) - 1)

Expand

(- 1 + sec^{2} (t)) sec^{2} (t) : - sec^{2} (t) + sec^{4} (t)

= - sec^{2} (t) + sec^{4} (t)

= sec^{4} (t) - sec^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

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

tan (A) = \frac{sec ( A )}{csc ( A )}

csc (- x) = - csc (x)

sin (β) + cos (β) cot (β) = csc (β)

cos (\frac{3 π}{2} + x) = - cos (x)

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