How can this be simplified to the most basic form? Please include a step-by-step explanation.

sec^4(x) - tan^4(x)

1 Answer
Mar 1, 2018

\quad "An answer is:" \qquad \qquad \ sec^4(x) - tan^4(x) = \ 1 + 2 tan^2(x).

Explanation:

"We have:"

\qquad sec^4(x) - tan^4(x)

\qquad \qquad = \ [ sec^2(x) ]^2 - [ tan^2(x) ]^2;

\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ color{blue}{"apply difference of two squares" \quad rArr }

\qquad \qquad = \ [ sec^2(x) - tan^2(x) ] cdot [ sec^2(x) + tan^2(x) ];

\qquad \qquad \qquad \qquad \quad color{blue}{"apply twice, trig identity:" \qquad 1 + tan^2x = sec^2x \quad rArr }

\qquad \quad = \ [ ( 1 + tan^2(x) ) - tan^2(x) ] cdot [ ( 1 + tan^2(x) ) + tan^2(x) ];

\qquad \qquad = \ [ 1 + tan^2(x) - tan^2(x) ] cdot [ 1 + tan^2(x) + tan^2(x) ];

\qquad \qquad = \ [ 1 + color{red}cancel{ tan^2(x) - tan^2(x) } ] cdot [ 1 + 2 tan^2(x) ];

\qquad \qquad = \ 1 cdot [ 1 + 2 tan^2(x) ];

\qquad \qquad = \ 1 + 2 tan^2(x).

"This an answer simplified to a basic form -- always, there may"
"be other basic forms to which it can be simplified, as well; but"
"this is one."

"Summary:"

\qquad \qquad \qquad \qquad \qquad \qquad sec^4(x) - tan^4(x) = \ 1 + 2 tan^2(x).