Given:
The trigonometric function is given as,
[tex]\sec ^4x-\tan ^4x[/tex]
The objective is to simplify the expression.
Explanation:
The given expression can be written as,
[tex]\sec ^4x-\tan ^4x=(\sec ^2x)^2-(\tan ^2x)^2\text{ . . . . . (1)}[/tex]
Now, consider the algebraic identity,
[tex]a^2-b^2=(a+b)(a-b)[/tex]
Then, the equation (1) can be written using the algebraic identity as,
[tex](\sec ^2x)^2-(\tan ^2x)^2=(\sec ^2x+\tan ^2x)(\sec ^2x-tan^2x)\text{ . . . (2)}[/tex]
Consider the Pythagorean identity,
[tex]\sec ^2x-\tan ^2x=1[/tex]
Then, equation (2) can be written as,
[tex]\begin{gathered} =(\sec ^2x+\tan ^2x)(1) \\ =(\sec ^2x+\tan ^2x)\text{ . . . . . (3)} \end{gathered}[/tex]
Hence, the simplified expression is sec²x + tan²x.
